Louise Goupil
email: lougoupil@gmail.com

Petter Johansson

Lars Hall

Jean-Julien Aucouturier

Vocal signals only impact speakers' own emotions when they are self-attributed

Keywords:

come     Vocal signals only impact speakers' own emotions when they are self-attributed

Introduction

Emotions are multidimensional states, with both physiological (e.g., increased breathing and heart rate), expressive (e.g., smiles, frowns, growls) and subjective components (e.g., happiness, sadness, fear). While this conceptualization of emotions is relatively uncontroversial, the order in which each of these components occur within an emotional episode still remains a matter of debate. Contrary to the common intuition that bodily changes follow subjective feelings, psychologists have long hypothesized that in fact, feelings result from the perception of physiological changes that arise in the body in response to an event [START_REF] James | The Principles of Psychology[END_REF][START_REF] Kuznetsova | Bodily influences on emotional feelings: Accumulating evidence and extensions of William James's theory of emotion[END_REF][START_REF] Schachter | Cognitive, social, and physiological determinants of emotional state[END_REF]. As William James famously stated, although "common-sense says, we loose our fortune, are sorry and weep, […] the more rational statement is that we feel sorry because we cry" [START_REF] James | The Principles of Psychology[END_REF]. While they differ in their formalization of how afferent bodily signals are integrated by the brain to give rise to conscious emotional experiences, most modern theories of emotions still highlight this as an important aspect of emotional experience [START_REF] Barrett | The theory of constructed emotion: An active inference account of interoception and categorization[END_REF][START_REF] Damasio | Feelings of emotion and the self[END_REF][START_REF] Ledoux | A higher-order theory of emotional consciousness[END_REF][START_REF] Panksepp | Neurologizing the psychology of affects: How appraisal-based constructivism and basic emotion theory can coexist[END_REF]. Yet, although a lot of research has been dedicated to this issue, it has proven difficult to empirically confirm whether or not the perception of bodily changes plays a causal role in the emergence of emotional subjective experience.

The main strategy has been to causally manipulate bodily signals [START_REF] Critchley | Interoception and emotion[END_REF][START_REF] Kuznetsova | Bodily influences on emotional feelings: Accumulating evidence and extensions of William James's theory of emotion[END_REF][START_REF] Schachter | Cognitive, social, and physiological determinants of emotional state[END_REF][START_REF] Strack | Inhibiting and facilitating conditions of the human smile: A nonobtrusive test of the facial feedback hypothesis[END_REF][START_REF] Wagenmakers | Registered replication report: Strack, Martin, & Stepper[END_REF]. The brain receives bodily signals through two main channels: either exteroceptively (tactile, visual or auditory inputs) or interoceptively (cardiac, intestinal or respiratory inputs), and evidence of acoustic compensation [START_REF] Aucouturier | Covert digital manipulation of vocal emotion alter speakers' emotional states in a congruent direction[END_REF]. Here, we rely on a more demanding procedure, involving frequent reversals of the vocal feedback manipulation in a within-participant paradigm. This new procedure induces a greater rate of explicit and implicit detections of the experimental manipulation, and notably allows us to detect possible acoustic compensation at the level of the participant. Using this new procedure, we asked participants to deliberate out loud about how they would feel in familiar scenarios that systematically varied in valence and arousal (e.g., receiving good or bad news). While the participants spoke, we covertly manipulated subtle acoustic features (fundamental frequency and timbre) in their voices in order to introduce affective signals typically associated with happiness or sadness (Fig. 1). We predicted that covert manipulation of affective signals in their own voice should impact participants' emotional experience in the congruent direction (e.g., happy signals should induce positive feelings). Additionally, we predicted that this effect should disappear when speakers detected our manipulation, either explicitly (as assessed by progressive questions during debriefing) or implicitly (as evidenced through vocal compensation).

Materials and methods

Experimental design

In order to assess the impact of self-attributed vocal affective signals on emotional experience, participants read 36 emotional scenarios out loud, and then deliberated out loud about how they think they would feel in this situation. After reading and describing how they would feel (which took on average 79 s ± 42 SD), participants had to summarize their assumed emotional state using: 1) a continuous valence scale ranging from "very negative" (0) to "very positive" (100); 2) a continuous arousal scale ranging from "not aroused" (0) at all to "very aroused" (100), and 3) to report how sure they were that they would feel this way on a continuous confidence scale ranging from "not confident at all" (0) to "very confident" (100). While they spoke, participants' voices were covertly manipulated to make them sound happy (i.e., higher fundamental frequency, and boost in high-medium frequencies mimicking the acoustic consequences of smiling, [START_REF] Ponsot | Uncovering mental representations of smiled speech using reverse correlation[END_REF] in 1/3 of the trials, sad (i.e., lower fundamental frequency, darker spectrum) in 1/3 of the trials, or remained unchanged (i.e., neutral condition) in the remaining 1/3 of the trials. The covert manipulation started as soon as participants began reading the scenario out loud, and increased progressively with a ramp of 2.5 s to reach pre-defined values for each acoustic manipulation (fundamental frequency and vocal timbre). The manipulation stayed on while the participants discussed out loud how they think they would feel in the scenario. Fig. 1. Experimental paradigm. Participants read vignettes depicting familiar emotional scenarios, and then deliberated out loud about how they would feel in these situations. They then summarized their emotional state on a valence scale, an arousal scale and a confidence scale. While they spoke, their voice was covertly manipulated to make it sound happy (i.e., higher fundamental frequency, brighter spectrum) or sad (i.e., lower fundamental frequency, darker spectrum), or remained unchanged (i.e., neutral).

Participants were fitted with closed headsets minimizing the contamination from environmental noise and their non-manipulated voice (Beyerdynamics -DT770), and their voice was recorded with a DPA 4088 headset and an RME UCX Fireface sound card, that allows for a roundtrip latency under 20 ms. An Apple MacBook Pro running PsychoPy [START_REF] Peirce | PsychoPy-Psychophysics software in Python[END_REF] was used to control stimulus presentation, recording responses, as well as communicate in real time with the voice transformation software (DAVID) to apply the various voice transformations depending on experimental conditions. Participants were asked to read each of the scenarios out loud, before speaking freely to describe how they would feel in such a situation. To conceal the true nature of the experiment, participants were told that the experiment aimed at uncovering why and how emotional feelings arise in various imaginary situations.

In a previous study using a between-subject design (i.e., each participant heard her/his voice manipulated in only one direction), and a very slow introduction of the vocal-feedback manipulation, we found both that most participants did not detect that their voice had been manipulated, and that none of the participants displayed acoustic compensation [START_REF] Aucouturier | Covert digital manipulation of vocal emotion alter speakers' emotional states in a congruent direction[END_REF]. In the present experiment, we used a design that we deemed likely to result in a higher detection rate, either explicitly or as indexed through acoustic compensation. This time, we used a within-subject design, in which the direction of the vocal-feedback manipulation was changed after every 4 trials, resulting in 9 blocks of 4 happy, sad or neutral trials. Thus, there were 3 non-consecutive blocks of 4 trials for each of the vocal feedback manipulation (hereafter VFM) conditions (12 trials with the happy VFM, 12 with the sad VFM, and 12 neutral trials). Pilot testing revealed that in this within-subject design, the percentage of participants who detected the manipulation increased as opposed to a between-subject design where only one effect was slowly applied to each participants' voice [START_REF] Aucouturier | Covert digital manipulation of vocal emotion alter speakers' emotional states in a congruent direction[END_REF]. Vocal feedback manipulation, block order and scenarios were pseudo-randomized across participants, such that across groups of 18 participants, each scenario appeared in the happy, sad or neutral condition for 1/3 of the participants.

Following the experiment, participants were debriefed with progressive questions, in order to assess whether they detected the vocal feedback manipulation or not. Depending on their responses, they were given a score of detection ranging from 1 to 6 by the experimenter (1 = "you intentionally manipulated my voice"; 2 = "my voice was higher / lower sometimes, and it changed during the experiment"; 3 = "my voice sounded strange sometimes, and it changed during the experiment"; 4 = "my voice sounded strange, and it was not only because I am not used to hearing myself through headphones"; 5 = "my voice sounded strange, but it is probably because I am not used to hearing myself through headphones" ; 6 = "my voice was absolutely fine"). At this stage, the experimenter remained blind to the participant's behavior during the experiment (i.e., did not hear the participant's responses during the experiment, did not know whether the participant compensated acoustically or not, and did not know whether they displayed the emotional effect or not). Participants also filled in a questionnaire in order to assess their level of alexithymia (TAS scale, Zimmermann, Quartier, Bernard, Salamin, & Maggiori, 2007) as well as social anhedonia (SHAPS scale, [START_REF] Gaillard | L'anhédonie dans la dépression[END_REF]. At the end of the experiment, they were debriefed and informed of the true purpose of the study.

Participants

Sample size was chosen following pilot testing, which suggested that approximately one third of the participants detected the manipulation explicitly, one third implicitly, and one third showed no detection at all. Given our design which required forming groups of 18 participants (see experimental design above), we thus aimed to test fifty-four participants. In total, fifty-five participants were tested, but seven participants had to be excluded due to technical problems with recording, or real-time vocal transformations (i.e., failure of the software, or communication between the software and the experimental interface coded in python), leaving N = 48 participants in the final sample (29 females; age = 23 ± 3.24 SD).

Stimuli

36 emotional scenarios were adapted and translated into French from (Wilson-Mendenhall, [START_REF] Wilson-Mendenhall | Neural evidence that human emotions share core affective properties[END_REF]. These scenarios were originally designed to describe familiar situations able to induce varying experience of valence and arousal in participants. We selected 9 scenarios per valence (positive/negative) and arousal (high/low arousal) quadrants, based on the normative ratings provided by [START_REF] Wilson-Mendenhall | Neural evidence that human emotions share core affective properties[END_REF] (see supplementary material for four examples of scenarios). We verified in an independent group of 10 participants with equivalent demographic background as our main sample (4 females, age = 28.5 ± 4.48 SD) that the stimuli were perceived similarly in our population (mean valence ratings for positive scenarios on a scale from 0 to 100: M = 69.7 ± 9 SD, negative scenarios: M = 40.2 ± 9.3 SD; arousal ratings for high arousal scenarios on a scale from 0 to 100: M = 60.3 ± 19 SD; low arousal scenarios: M = 48.5 ± 11.1 SD).

Voice transformation

Subtle affective signals were artificially introduced in participants' speech in real time using the voice transformation technique introduced in [START_REF] Rachman | DAVID: An open-source platform for real-time transformation of infrasegmental emotional cues in running speech[END_REF]. The software, called DAVID, uses a selection of digital audio effects such as fundamental frequency shifting and spectral filtering to simulate emotional expression in running speech, with a very-low latency compatible with real-time vocal production. For the happy Vocal Feedback Manipulation (VFM), we applied a positive fundamental frequency shift (+50 cents) as well as a spectral modification aiming to simulate the impact of smiling (notch filter at 2880 Hz, gain = 3, Q = 0.74) [START_REF] Ponsot | Uncovering mental representations of smiled speech using reverse correlation[END_REF]. For the sad VFM, we applied a negative fundamental frequency shift (-70 cents) as well as a spectral modification aiming to attenuate the power in high frequencies, resulting in a darker sound (high-shelf filter, 8000 Hz, gain = 0.25, Q = 1) [START_REF] Aucouturier | Covert digital manipulation of vocal emotion alter speakers' emotional states in a congruent direction[END_REF][START_REF] Scherer | Vocal communication of emotion: A review of research paradigms[END_REF]. For the neutral condition, the voice was routed through the same algorithm so that the latency (20 ms) and processing of the voice was the same as in the VFM conditions, except that no transformation was applied.

Pre-processing and statistical analysis

For each trial and each participant, we used the Praat software [START_REF] Boersma | Praat, a system for doing phonetics by computer[END_REF] to compute the fundamental frequency from two different recordings of the participant's voice: the non-modified recording (i.e., the natural voice produced by the participant, which is the input of the voice transformation algorithm) and the modified recording (i.e., the artificial voice heard by the participant, which is the output of the voice transformation algorithm). This analysis revealed that, although generally effective, our fundamental frequency transposition algorithm sometimes failed because of the low vocal or audio quality in certain trials (e.g., because the position of the microphone was not adequate). Thus, we excluded trials for which the difference between the fundamental frequency of the modified and non-modified voice did not show the intended transposition (e.g., for the sad condition, we excluded trials if the fundamental frequency of the modified voice was not lower than the fundamental frequency of the non-modified voice). This pre-processing lead to excluding 7% of the data.

Trials in the neutral VFM condition were then used as a baseline to assess the impact of the happy and sad VFM on ratings. For each participant, ratings were normalized with respect to the participant's neutral trials, following this formula: zX = (X -m_neutral)/ sd_neutral, were zX corresponds to normalized ratings (valence, arousal or confidence), X to data samples, m_neutral to the average and sd_neutral to the standard deviation of the ratings given by this participant in neutral trials. Over all participants, the average neutral ratings were: valence: 52.95 ± 7.47 SD; arousal: 52.01 ± 8.13 SD; confidence: 79.42 ± 13.05 SD.

Explicit detection was assessed from participants' responses during the debriefing: participants who detected that their voice had been manipulated (i.e., detection scores <= 4) constituted the group of explicit detectors, and participants who did not detect the manipulation at all (scores > 4) constituted the group of non-detectors (see Table 1).

Implicit detection was estimated for each participant by assessing whether the fundamental frequency of their non-modified voice deviated from its normal range in each of the two VFM conditions (happy and sad). We assessed acoustic compensation at the level of the participant (i.e. globally, over all trials of the participant) by testing whether the fundamental frequency extracted from the nonmodified recordings in each of the VFM condition (happy / sad) differed from the normal range of variation in the participant's fundamental frequency, measured across all conditions. The fundamental frequency of the non-modified recordings of each participant was normalized (z-scored) over all trials of all three conditions of the participant. The z-scored values for the happy and sad trials of the participant were then tested against zero, with two separate one-sample t-tests, to assess their deviation from the participant's normal range. A significance threshold of <0.1 was adopted in order to avoid missing implicit detection through lack of power at the individual level. Similar results were obtained when using a threshold of p < 0.05 to classify individual participants, and also when normalizing with respect to neutral trials only.

This analysis revealed that 50% (N = 24; 16 explicit non-detectors) of the participants did not show any acoustic compensation; 16.7% (N = 8; 5 non-detectors) compensated in the happy condition (i.e., produced a lower fundamental frequency when their voice had been transposed upwards); 22.9% (N = 11; 7 non-detectors) compensated in the sad condition (i.e., produced a higher fundamental frequency when their voice had been transposed downward); 2% (N = 1 non-detector) compensated in both directions; unexpectedly, 8.3% (N = 4; 2 non-detectors) of the participants displayed anti-compensation in the happy condition (i.e. their voice had a higher than average fundamental frequency in the happy trials).

For each individual participant and condition, we classified all trials of a condition as "compensated" if the participant compensated in that condition (conservatively, the anti-compensations observed in the happy condition were also counted as compensation). This resulted in 188 trials (96 for happy, 92 for sad) in the compensated condition, and 506 trials (276 for happy, 230 for sad) in the noncompensated condition. Fig. S1 shows an analysis of the fundamental frequency of the non-modified voices for compensated versus non-compensated trials, which confirmed that this classification efficiently separated trials with substantial acoustic compensation from trials without acoustic compensation at the level of the group.

Because of the unbalanced numbers of detectors vs. non-detectors, and compensated vs. non-compensated trials, we used hierarchical linear mixed regressions to assess the statistical significance of our results. Hierarchical linear models were fit with the maximum likelihood method. Normalized valence ratings, confidence ratings, arousal ratings, the non-modified (pre-VFM) and the modified (post-FVM) fundamental frequency served as dependent variables in separate models. Participant and sentence number were entered as random factors. Fixed factors were the VFM condition (happy vs. sad vs. neutral), the explicit detection condition (detector vs. non-detector as between-participant factor, determined as described above), the detection level (using each of the six levels of the detection scale), or acoustic compensation (compensated vs. not-compensated trials, as described above). We report beta estimates, standard errors, t-values (with Satterthwaite approximations to degrees of freedom), p-values and chi-squares for hierarchical nested model comparisons with likelihood ratio tests [START_REF] Gelman | Data analysis using regression and multilevel/hierarchical models[END_REF] with the lme4 and lmerTest packages in R (Kuznetsova et al

Results

Impact of participants' felt emotions on the fundamental frequency of their voice

Before assessing the impact of the Vocal Feedback Manipulation (hereafter VFM) on participants' emotions, we first examined how the fundamental frequency of participants' non-modified voices varied with their emotional reports in the non-manipulated trials (i.e., the neutral condition). This procedure allowed us to check that we could replicate previous findings concerning the relationship between the fundamental frequency of speakers' voices, and their self-reported valence and arousal [START_REF] Aucouturier | Covert digital manipulation of vocal emotion alter speakers' emotional states in a congruent direction[END_REF][START_REF] Scherer | Vocal communication of emotion: A review of research paradigms[END_REF]. More importantly, it also allowed us to examine the relationship between fundamental frequency and confidence in emotional judgements. Previous research has documented that epistemic uncertainty (i.e., doubting about one's own state of knowledge) is typically associated with a rising intonation and reduced volume [START_REF] Goupil | Event-related prosody reveals distinct acoustic manifestations of accuracy and confidence in speech[END_REF][START_REF] Jiang | The sound of confidence and doubt[END_REF], but the acoustic correlates of emotional uncertainty (i.e., doubting about one's own emotional state) remain unclear so far.

The fundamental frequency of participants' voices in neutral trials was averaged separately depending on whether participants reported feeling a negative or positive emotion (median split of valence ratings, see Fig. 2A), high or low arousal (Fig. 2B) and high or low confidence (Fig. 2C). Trials in which participants reported negative emotions were accompanied by lower fundamental frequency than trials in which they reported positive emotions (t(47) = 2.17, p = 0.035, d = 0.44). Similarly, trials with low confidence on the reported emotion had lower fundamental frequency than trials with high confidence (t(46) = 2.8, p = 0.007, d = 0.51). However, arousal only marginally impacted fundamental frequency, with low arousal corresponding to a lower fundamental frequency than high arousal (t(47) = 1.96, p = 0.056, d = 0.37).

Impact of the vocal feedback manipulations on participants' felt emotionswhole group

Next, we turned to our main research question, and examined whether the VFM had a main effect on the valence, arousal and confidence ratings provided by the participants, and whether this effect was congruent with the relationship between participants' felt emotions and the fundamental frequency of their voice reported above. Over the whole group of participants (detectors and nondetectors, compensators and non-compensators), a repeated measure multi-variate analysis of variance (rm-MANOVA) revealed a significant effect of the VFM condition (happy / neutral / sad) on our three main measures of valence, arousal and confidence ratings (F (2,94) = 2.89, Pillai's trace = 0.17, p = 0.01). This first analysis at the level of the whole group of participants, and over the three dimensions of emotional feelings explored in this study, is compatible with our first hypothesis that VFM should impact emotional feelings.

Impact of explicit detection on the vocal feedback emotional effect

We assessed explicit detection through a progressive debriefing at the end of the experiment (see methods for details). As can be seen in Table 1, out of the 48 participants, 31 (65%) did not detect the manipulation explicitly (i.e., detection score of 6 "my voice was absolutely fine": N = 23 ; or 5 "my voice sounded strange, but it is probably because I am not used to hearing myself through headphones": N = 8), and 17 (35%) participants detected that their voice had been manipulated (i.e., detection score of 4 "my voice sounded strange, and it was not only because I am not used to hearing myself through headphones": N = 1; 2 "my voice was higher / lower sometimes, and it changed during the experiment": N = 14; or 1 "you intentionally manipulated my voice": N = 2). Of note, 23 participants reported that "their voice was absolutely fine", even though they were given the opportunity to respond "my voice sounded strange, but it is probably because I am not used to hearing myself through headphones", an option chosen by a comparatively low number of participants (N = 8). This indicates that roughly half or the participants had no problems with self-identifying their voice in this experiment, despite the fact that it was heard through headphones, which necessarily transforms the vocal feedback they normally hear through the combination of air and bone conduction [START_REF] Pörschmann | Influences of bone conduction and air conduction on the sound of one's own voice[END_REF].

Crucially, and consistent with our second hypothesis, the main effect of the VFM on valence, confidence and arousal reports was driven by the 31 participants that did not explicitly detect the vocal feedback manipulation (see Fig. 3). The interaction between explicit detection (yes/no) and the VFM condition (happy/neutral/sad) was tested on our three measures separately (valence, confidence and arousal normalized ratings), in hierarchical linear mixed regressions analyses including participant number and sentence number as random factors.

For valence ratings, there was no main effect of the VFM condition (X2 = 1.47, p = 0.48), but an interaction between the VFM and explicit detection (X2 = 7.3, p = 0.026, see Fig. 3A). This interaction reflected the fact that in the group of non-detectors, the happy VFM significantly increased valence ratings compared to the baseline (i.e., neutral) condition (M = 0.16 ± 0.3 SD; 95% CI [0.04, 0.27]; beta = 0.15 ± 0.06 se, t = 2.5, p = 0.014, d = 0.51), and marginally so compared to the sad condition (M = -0.02 ± 0.35 SD; 95% CI [-0.15, 0.1]; beta = 0.11 ± 0.06 se, t = 1.8, p = 0.07, d = 0.56, the difference between the sad VFM and the neutral condition was not significant, beta = -0.04 ± 0.06 se, t = 0.58, p = 0.56, d = 0.07). In the group of detectors, none of these comparisons were significant (all p-values > 0.14 and d < 0.3). In addition, valence ratings in the happy condition were higher in the group of non-detectors than in the group of detectors (beta = 0.27, se = 0.08, t = 3.2, p = 0.002, d = 0.83). The direction of the effect is consistent with the fact that speakers displayed higher fundamental frequency when they reported feeling positive rather than negative emotions (see Fig. 2A, for the whole group of participants). As can be seen in Fig. 3B, the interaction between the impact of the VFM on valence ratings and explicit detection was also observable when examining detection as a continuous variable: there was a significant interaction between the VFM and detection level (from 1: highest detection to 6: lowest level of detection; X2 = 6.42, p = 0.04). The difference between the happy condition and the neutral condition increased with detection level (beta = 0.07, se = 0.03, t = 2.5, p = 0.012), but this effect was not significant for the sad condition (beta = 0.026, se = 0.027, t = 0.96, p = 0.33).

Regarding confidence, there was a significant effect of the VFM (X2 = 8.52, p = 0.014, see Fig. 3C) and no significant interaction between the VFM and explicit detection (X2 = 1.07, p = 0.58). The sad VFM significantly decreased confidence ratings (M = -0.27 ± 0.73 SD; 95% CI [-0.54, -0.001]) compared to both neutral (beta = -0.3 ± 0.1 se, t = -2.8, p = 0.005, d = 0.37) and happy VFM conditions (M = -0.06 ± 0.7 SD; 95% CI [-0.32, 0.20], beta = -0.25 ± 0. 11 se, t = -2.28, p = 0.02, d = 0.3) in the group of explicit non-detectors. In the group of explicit detectors, none of the comparisons were significant (all p-values > 0.39 and d < 0.13; all other comparisons were non-significant). Again, the direction of the effect was consistent with the fact that participants spoke with a lower fundamental frequency when they reported lower confidence (see Fig. 2C, for the whole group of participants).

Regarding arousal, there was a main effect of the VFM (X2 = 7.9, p = 0.019) and no interaction between the VFM and explicit detection (X2 = 0.25, p = 0.9). As can be seen in Fig. 3D, there was a non-specific effect of the vocal feedback manipulation where both the sad (M = 0.11 ± 0.5 SD; 95% CI [0, 0.25]; beta = 0.13 ± 0. 07 se, t = 1.88, p = 0.059, d = 0.22) and the happy VFM (M = 0.12 ± 0.4 SD; 95% CI [0, 0.23]; beta = 0.16 ± 0. 07 se, t = 2.2, p = 0.027, d = 0.3) increased arousal ratings compared to neutral trial), with no difference between the happy and the sad conditions (beta = 0.03 ± 0. 07 se, t = 0.4, p = 0.69, d = 0.15). Thus, regardless of the specific manipulation, manipulated trials led to higher arousal ratings. We come back to this aspect in the discussion.

In sum, explicit detection of the vocal feedback manipulation generally removed its impact on emotional self-reports for valence and confidence, which were also the two emotional dimensions that were more tightly associated with changes in the fundamental frequency of participants voices in non-modified trials. This finding suggests that affective vocal signals have to be perceived as selforiginated in order to directly influence feelings. To further examine this hypothesis, we then turned to our marker of 'implicit' error detection in speech production, by examining acoustic compensation in the subgroup of 31 (explicit) non-detectors.

Implicit detection of the VFM indexed by acoustic compensation

At the level of the group, we observed substantial acoustic compensation, with a significant main effect of the VFM on the fundamental frequency of the voice produced in manipulated trials (X2 = 44, p < 0.001) and no significant interaction between the VFM and explicit detection (X2 = 4.2, p = 0.12). To document the specific aspect of implicit detection, over and beyond explicit detection, we now focus on non-detectors only (see Fig. S1A for details in the group of explicit detectors). As can be seen in Fig. S1A, the happy VFM lead the non-detectors to lower their voice compared to neutral trials (M = -0.20 ± 0.36; 95% CI [-0.36, -0.001]; beta = 0.18 ± 0.07 se, t = 2.6, p = 0.009, d = 0.57), while the sad VFM lead them to raise their voice (M = 0.27 ± 0.45; 95% CI [0.1, 0.44]; beta = -0.29 ± 0.073 se, t = 3.96, p < 0.001, d = 0.6). At the individual level, we found that out of the 31 participants who failed to explicitly detect the VFM, 15 of them (31% of the total sample of N = 48) showed substantial acoustic compensation for one or both of the VFM conditions (i.e., the fundamental frequency of their voice deviated from their usual range when the happy or sad VFM was applied, see methods for details). For each participant, trials in a given condition were classified into two "implicit detection" conditions: compensated if the participant showed substantial acoustic compensation for that condition over the whole experiment, and non-compensated if the participant did not show acoustic compensation for that condition over the whole experiment.

Impact of implicit detection on the vocal feedback emotional effect

To assess whether implicit detection impacted the effect of the VFM, we ran hierarchical linear mixed regressions on our three measures separately (valence/confidence/arousal normalized ratings) with the VFM condition (happy/neutral/sad) and acoustic compensation (yes/no) as fixed factors, and participant number and sentence number as random factors.

Regarding valence ratings (Fig. 4A; also see Fig. S2A for histograms allowing to visualize the distribution of the data), there was a marginal effect of the VFM condition (X2 = 5.17, p = 0.075), no main effect of compensation (X2 = 2.47, p = 0.11) and an interaction between compensation and the VFM (X2 = 6, p = 0.047) reflecting the fact that the impact of the VFM on valence ratings was limited to non-compensated trials. In non-compensated trials, there was a significant increase in valence ratings in the happy VFM condition (M = 0.25 ± 0.27 SD) compared to both neutral (beta = 0.25 ± 0.094 se, t = 2.6, p = 0.009, d = 0.93) and sad trials (M = -0.04 ± 0.37 SD, beta = 0.22 ± 0.089 se, t = 2.53, p = 0.012, d = 0.92). Ratings in the sad condition did not significantly differ from the neutral condition (beta = -0.02 ± 0.1 se, t = -0.2, p = 0.83, d = 0.12). By contrast, none of these comparisons were significant in compensated trials (happy M = -0.1 ± 0.26 SD, vs. neutral: beta = -0.1 ± 0.12 se, t = -0.8, p = 0.43, d = 0.38; happy vs. sad M = 0.04 ± 0.26 SD, beta = -0.135 ± 0.15 se, t = -0.93, p = 0.35, d = 0.58; sad vs. neutral: beta = 0.03 ± 0.13 se, t = -0.27, p = 0.8, d = 0.15).

Similarly, for confidence ratings there was a significant effect of the VFM (X2 = 10.17, p = 0.006), no main effect of acoustic compensation (X2 = 0.15, p = 0.7), and an interaction between compensation and the VFM (X2 = 5.94, p = 0.05), reflecting the fact that compensation also abolished the impact of the VFM on confidence ratings (see Fig. 4B and Fig. S2B for histograms). In noncompensated trials, there was a significant decrease in confidence ratings in the sad VFM condition (M = -0.33 ± 0.82 SD) compared to both neutral (beta = -0.36 ± 0.14 se, t = -2.6, p = 0.009, d = 0.4) and happy trials (M = 0.04 ± 0.54 SD, beta = -0.43 ± 0.13 se, t = -3.4, p < 0.001, d = 0.54). Ratings in the happy condition did not significantly differ from the neutral condition (beta = 0.07 ± 0.13 se, t = 0.52, p = 0.6, d = 0.07) trials. None of these comparisons were significant in compensated trials (happy M = -0.34 ± 0.97 vs. neutral: beta = -0.34 ± 0.18 se, t = -1.9, p = 0.054, d = 0.35; happy vs. sad, M = -0.11 ± 0.25: beta = -0.21 ± 0.22 se, t = -0.96, p = 0.33, d = 0.35; sad vs. neutral: beta = -0.13 ± 0.18 se, t = -0.72, p = 0.5, d = 0.44).

Finally, regarding arousal ratings, there was no interaction between the VFM and acoustic compensation (X2 = 2, p = 0.36, see Fig. 4C), confirming the non-specificity of the impact of the VFM on arousal ratings.

In sum, as was also the case for explicit detection, implicit detection abolished the impact of the VFM on valence and confidence ratings. Importantly, the disappearance of the effect in compensated trials was not trivially due to acoustic compensation somehow counterbalancing the vocal transformations: the fundamental frequency of modified voices in compensated trials still largely reflected the intended transposition in this subset of the data (see Fig. 4D). In compensated trials, the fundamental frequency of the manipulated voices in the happy VFM condition (z-scored values, M = 1.19 ± 0.06 SD) remained, as intended, significantly higher than the fundamental frequency of neutral voices (M = -0.13 ± 0.19 SD, beta = 1.33 ± 0.05 se, t = 27, p < 0.001) and sad manipulated voices (M = -1.08 ± 0.12 SD, beta = 2.3 ± 0.06 se, t = 40, p < 0.001), and the fundamental frequency of the manipulated voices in the sad VFM condition remained significantly lower than the fundamental frequency of neutral voices (beta = -0.95 ± 0.05 se, t = 19, p < 0.001). This was similar to the pattern observed in non-compensated trials: the fundamental frequency of the manipulated voices in the happy VFM condition was significantly higher than the fundamental frequency of neutral voices (beta = 1.15 ± 0.04 se, t = 31, p < 0.001), and sad manipulated voices (beta = 2.22 ± 0.03 se, t = 64, p < 0.001), and the fundamental frequency of the manipulated voices in the sad VFM condition remained significantly lower than the fundamental frequency of neutral voices (beta = 1.06 ± 0.04 se, t = 28, p < 0.001). Thus, the impact of the VFM was not abolished in compensators because acoustic compensation canceled our experimental manipulation. Rather, we suggest that acoustic compensation reflected an implicit detection that the affective signals perceived in the voice were not self-generated, leading to a dismissal of these signals during the construction of emotional judgement.

Interaction between the vocal feedback effect and the background valence of the scenarios

Finally, we explored whether the impact of the VFM varied depending on the background valence of the scenarios (Fig. 5). Given Fig. 5. Interaction between the impact of the VFM on valence ratings and valence of the context. The percentage of negative responses given by each participant (as defined by a median split of valence ratings) was computed separately in each of the VFM conditions and Valence of the Scenario (green: happy, grey: neutral; blue: sad); * p < 0.05; ** p < 0.01. Comparisons across the positive/negative dimensions are not shown for clarity, but all of them were highly significant (p-values < 0.01). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) that we did not replicate the previous finding that a sad VFM should decrease valence ratings, we were particularly interested in whether the valence of the context may impact this effect [START_REF] Aucouturier | Covert digital manipulation of vocal emotion alter speakers' emotional states in a congruent direction[END_REF]

employed a single, neutral text stimulus).

To analyze whether the effect of the VFM depends on the valence of the scenario, we computed for each participant, scenario type (positive/negative) and VFM (happy, neutral, sad) the probability of giving a negative response (determined by a median split on the set of all valence ratings given by each participant). A linear mixed regression including the percentage of negative responses as a dependent variable, the valence of the scenario and the VFM as fixed factors, and participant as a random factor revealed, expectedly, a main effect of the valence of the scenario (X2 = 51.4, p < 0.001), but also an interaction between the VFM and the valence of the scenario (X2 = 9.87, p = 0.0072). Although for positive scenarios the sad VFM (M = 0.43 ± 0.2 SD) increased the percentage of negative responses compared to neutral trials (M = 0.32 ± 0.12 SD, beta = 0.13 ± 0.05 se, t = 2.75, p = 0.007, d = 0.71) and happy trials (M = 0.33 ± 0.17 SD, beta = 0.12 ± 0.04 se, t = 2.84, p = 0.005, d = 0.56), for negative scenarios (M = 0.67 ± 0.23 SD) it did not significantly impact the percentage of negative responses compared to neutral trials (M = 0.72 ± 0.12 SD, beta = -0.05 ± 0.04 se, t = -1.1, p = 0.28, d = 0.26), and happy trials (M = 0.64 ± 0.17 SD, beta = 0.03 ± 0.037 se, t = -0.89, p = 0.37, d = 0.16). Conversely, the happy effect decreased the percentage of negative responses compared to the neutral condition for negative scenarios (beta = -0.08 ± 0.04 se, t = -2.1, p = 0.04, d = 0.55), but not significantly so in positive scenarios (beta = 0.009 ± 0.04 se, t = -0.2, p = 0.83, d = 0.08). In sum, the happy VFM mostly impacted judgements in negative scenarios, while the sad VFM mostly impacted judgements in positive scenarios.

By contrast, the valence of the scenario did not interact with the effect of the VFM on confidence ratings, in a similar analysis involving the percentage of confident responses (no interaction between the VFM and the valence of the scenario, X2 = 1.1, p > 0.5, see Fig. S3). This speaks against the possibility that the effect of the VFM on confidence ratings was due to perceiving a mismatch between the valence of the scenario and the sad vocal feedback effect.

Discussion

Covertly introducing artificial affective signals in their voice impacted participants' judgements about how they would feel in various emotional scenarios, but only when they did not detect the experimental manipulation. Such vocal signals are generally assumed to have evolved to serve a communicative function [START_REF] Dezecache | An evolutionary approach to emotional communication[END_REF][START_REF] Fitch | Calls out of chaos: The adaptive significance of nonlinear phenomena in mammalian vocal production[END_REF][START_REF] Scherer | Vocal communication of emotion: A review of research paradigms[END_REF], but here we find that they also play a role for the emergence of conscious emotional states at the individual level.

Applying the happy effect to participants' voices lead them to report more positive emotions. This result is consistent with our initial predictions and with previous research [START_REF] Aucouturier | Covert digital manipulation of vocal emotion alter speakers' emotional states in a congruent direction[END_REF]. By contrast, we did not observe an overall significant decrease in emotional valence in the sad condition. This contradicts our prediction that sad VFM should lead to a decrease in valence ratings, and no impact on arousal ratings [START_REF] Aucouturier | Covert digital manipulation of vocal emotion alter speakers' emotional states in a congruent direction[END_REF]. Several factors may explain the fact that the sad VFM did not significantly impact valence ratings here. In particular, it appears that the impact of the sad VFM on emotional judgements was affected by contextual information, and was only manifest in positive scenarios, an issue to which we come back below.

Regarding arousal, we observed a non-specific effect of the VFM, whereby both types of manipulated trials corresponded to higher ratings (Fig. 3). It is possible that, in the context of the present study (reading scenarios out loud), participants did not consistently associate higher or lower fundamental frequency to arousal. This would be consistent with the fact that, contrary to valence, arousal ratings were not strongly associated with the fundamental frequency of participants voices in non-manipulated (neutral) trials (Fig. 2). Instead of an emotional effect, it may have been possible that this increase in arousal ratings reflects physiological arousal linked to the detection of a fundamental frequency mismatch in speech production, regardless of the direction of the change. However, the fact that neither explicit or implicit detection interacted with the increase of arousal makes this possibility unlikely in our view.

The present findings partially replicate a previous study [START_REF] Aucouturier | Covert digital manipulation of vocal emotion alter speakers' emotional states in a congruent direction[END_REF], but also extend it in several ways, documenting complexities that have important theoretical and methodological implications. First, here we systematically varied the direction of manipulation of the participants' voice within-subject, while the initial study employed a single manipulation which varied betweensubjects. In this novel setting, the VFM is sometimes detected by the speakers, either implicitly or explicitly, which allowed us to examine the impact of the detection on the VFM effect. Crucially, we observed that the impact of vocal affective signals depended on these acoustic modifications being perceived as self-generated: although 65% of the participants remained unaware of the manipulation, the substantial number of participants who detected the VFM showed no emotional effect at the group level. We also found that implicit detection that the vocal signal is not self-generated, as defined by acoustic compensation of the manipulation in the participant's non-modified voice, also led to a dismissal of the affective information it contained.

Research has shown that perturbing speech can result in vocal compensation even when participants are not aware of the manipulation [START_REF] Hafke | Nonconscious control of fundamental voice frequency[END_REF][START_REF] Jones | Perceptual calibration of F0 production: Evidence from feedback perturbation[END_REF], and that this phenomenon is related to forward-models and self-identification [START_REF] Max | Sensorimotor adaptation to auditory perturbations during speech: Acoustic and kinematic experiments[END_REF][START_REF] Tian | Dynamics of self-monitoring and error detection in speech production: Evidence from mental imagery and MEG[END_REF]. In our initial study on the vocal feedback effect [START_REF] Aucouturier | Covert digital manipulation of vocal emotion alter speakers' emotional states in a congruent direction[END_REF], we did not observe any acoustic compensation. This is likely due to the fact that we used a continuous stimulation over a long period of time (12 min), and did not change the directionality of the VFM. Here, by contrast, using conditions that most closely resemble the conditions used in speech perturbation studies [START_REF] Hafke | Nonconscious control of fundamental voice frequency[END_REF][START_REF] Jones | Perceptual calibration of F0 production: Evidence from feedback perturbation[END_REF][START_REF] Max | Sensorimotor adaptation to auditory perturbations during speech: Acoustic and kinematic experiments[END_REF][START_REF] Tian | Dynamics of self-monitoring and error detection in speech production: Evidence from mental imagery and MEG[END_REF], we did find substantial acoustic compensation in a subset of our participants. Furthermore, we find that such implicit detection also removes the influence of affective vocal signals on emotional experience. Taken together with past research showing that acoustic compensation reflects the detection of a mismatch between the vocal feedback and internal predictions about the auditory consequences of planned articulation, and that this source monitoring process is linked to perceiving the vocal feedback as originating from the self or not [START_REF] Jones | Perceptual calibration of F0 production: Evidence from feedback perturbation[END_REF][START_REF] Tian | Dynamics of self-monitoring and error detection in speech production: Evidence from mental imagery and MEG[END_REF], our finding suggests that only those signals that are identified as originating from the self directly impact felt emotions. This indicates that the identification of bodily signals as originating from the self, potentially through a mechanism of internal forward models, is crucial for them to influence emotional feelings.

One possibility for future studies concerns the factors underlying the individual variability that we observed in detecting the VFM. Here, we did not find any strong association between levels of detection and individual variables such as gender (% of female explicit detectors: 37%; males: 32%), alexithymia (no differences between scores on the TAS scale between detectors and non-detectors, t(46) = -0.43, p = 0.7) or anhedonia (no differences between scores on the TAS scale between detectors and non-detectors, t(46) = 0.12, p = 0.9). Further research could further elucidate if there are other individual traits associated with detecting versus assimilating vocal feedback manipulations. We did, however, observe a correlation between alexithymia and the impact of the VFM on emotions (see Fig. S4). This effect is in line with studies reporting differences in the extent to which individuals are able to monitor their own interoceptive and bodily responses [START_REF] Critchley | Interoception and emotion[END_REF][START_REF] Kuznetsova | Bodily influences on emotional feelings: Accumulating evidence and extensions of William James's theory of emotion[END_REF], but would have to be confirmed in a bigger sample of participants.

We also observed that the background valence of the scenarios interacted with the VFM emotional effect, with the happy VFM mostly impacting negative scenarios, and the sad VFM mostly impacting positive scenarios. Several alternative interpretations can be proposed to account for this observation. First, it is important to note that although we labeled our effects happy and sad based on previous research on vocal affects [START_REF] Aucouturier | Covert digital manipulation of vocal emotion alter speakers' emotional states in a congruent direction[END_REF][START_REF] Scherer | Vocal communication of emotion: A review of research paradigms[END_REF], it is likely that the acoustic signals that we introduced here can actually be interpreted quite differently depending on the context. For instance, neutral states of low arousal such as boredom have also been associated with lower fundamental frequency, while highly negative states of activation such as anger tend to be associated with a higher fundamental frequency [START_REF] Scherer | Vocal communication of emotion: A review of research paradigms[END_REF]. Thus, one possibility is that the "happy" and "sad" effects actually lead to different inferences depending on the emotion that is afforded by a given scenario. Alternatively, given that we find that the happy VFM mostly impacts valence judgements in negative situations, while the sad VFM mostly impacts valence judgements in positive situations, it may be that congruent VFM (happy on happy, sad on sad) does not warrant an additive effect (for instance, because vocal tone is not informative or surprising in that context), while the same transformations carries more affective information in incongruent situations. The present experiment was not specifically designed to examine these interactions, so follow-up experiments that manipulate vocal signals and situational factors (i.e., whether the scenario depicts a scene that primarily affords joy, weariness, shame, …) orthogonally are required to better understand how the contextual information and vocal signals are integrated to give rise to emotions. Regardless of the precise mechanisms that are involved in these interactions, the present results suggest that, like the interpretation of emotional displays of others [START_REF] Crivelli | Facial Displays Are Tools for Social Influence[END_REF][START_REF] Wharton | Pragmatics and non-verbal communication[END_REF], the interpretation of self-generated signals is subject to an inferential process that integrates contextual cues. This would be consistent with theoretical frameworks proposing that emotional feelings result from integrative inferential processes that integrate bodily signals with other types of information such as episodic and semantic memories [START_REF] Barrett | The theory of constructed emotion: An active inference account of interoception and categorization[END_REF][START_REF] Ledoux | A higher-order theory of emotional consciousness[END_REF][START_REF] Schachter | Cognitive, social, and physiological determinants of emotional state[END_REF].

Another novel aspect of our results is that vocal feedback not only impacted emotional self-reports, but also, how confident participants were that they would feel this way. In neutral trials, we found a lower fundamental frequency in trials where participants provided lower confidence ratings (see Fig. 2C). Congruently, we also found that the sad VFM decreased confidence in emotional judgements, thereby suggesting that beyond impacting emotional judgements, the vocal feedback may also play a role in "metaemotional" judgements (i.e., judgements about emotional judgements). This is consistent with reports showing that confidence in perceptual judgements can be impacted by arousal [START_REF] Allen | Unexpected arousal modulates the influence of sensory noise on confidence[END_REF], and calls for the development of embodied models of metacognition. It would be especially interesting to test whether a similar vocal feedback effect extends to general epistemic judgements, e. g. using feedback with modified fundamental frequency to influence a participant's judgements of confidence in non-emotional tasks, such as how certain they are that they see a certain stimulus in a noisy visual search task (see also [START_REF] Goupil | Listeners' perceptions of certainty and honesty of another speaker are associated with a common prosodic signature[END_REF].

Our findings have several methodological implications. We have developed a paradigm that allows manipulating participants' vocal feedback in a within subject design, a methodological improvement compared to our previous study [START_REF] Aucouturier | Covert digital manipulation of vocal emotion alter speakers' emotional states in a congruent direction[END_REF]. This will enable further investigations, in particular to study the neural underpinnings of this vocal feedback emotional effect using traditional neuroimaging block paradigms. Our findings also delineate the conditions in which covertly manipulated vocal feedback can impact speakers' emotions: the emotional effect is present only when participants identify the vocal feedback as self-generated. We also observed that the background valence of the context substantially impacts the emotional effect. These things combined have implications for developing vocal feedback applications for clinical practice, like potential non-pharmacological treatments for posttraumatic stress disorder or preoperative anxiety [START_REF] Guerrier | Vocal markers of preoperative anxiety: a pilot study[END_REF][START_REF] Scherer | Investigating voice quality as a speaker-independent indicator of depression and PTSD[END_REF].

Conclusions

To conclude, the present study reveals that vocal self-perception plays a role in the emergence of emotional feelings, reflecting an inferential process integrating both the monitoring of physiological changes and the interpretation of contextual information. Our findings support theories that see perception and cognitive interpretation of bodily signals as central for the construction of idiosyncratic emotional experiences [START_REF] Barrett | The theory of constructed emotion: An active inference account of interoception and categorization[END_REF][START_REF] Damasio | The nature of feelings: Evolutionary and neurobiological origins[END_REF], and suggest that, beyond interoception, exteroception of self-originated signals also plays a substantial role.

CRediT authorship contribution statement

Fig. 2 .

 2 Fig. 2. Impact of felt emotions on the voice. Fundamental frequency of the voices in neutral (i.e., non-modified) trials depending on whether participants reported feeling A) a negative or positive emotion (median split); B) high or low arousal and C) high or low confidence (right panel). Error bars show 95% confidence intervals, and black asterisks the significance of paired t-tests comparisons with: . p < 0.1; * p < 0.05; ** p < 0.01.

Fig. 3 .

 3 Fig. 3. Main results depending on Explicit Detection. (A) Valence ratings (z-scored) depending on the vocal-feedback manipulation (VFM; happy: green, sad: blue) and explicit detection. Ratings were normalized with respect to neutral trials, so the neutral condition is not shown in the Figure (i. e., normalized ratings in the neutral condition were equal to zero). (B) Difference between valence ratings in the happy minus sad condition as a function of the level of detection. Dots show individual data. We show the best fitting regression line with 95% confidence intervals. (C) Confidence ratings (normalized with respect to neutral trials) depending on the VFM and explicit detection. (D) Arousal ratings (normalized with respect to neutral trials) depending on the VFM and explicit detection. Error bars represent 95% confidence intervals. White asterisks represent significance of the model comparisons between experimental conditions (happy and sad VFM) and the baseline condition (neutral), black asterisks represent significance of the model comparisons between the two experimental conditions within groups, and grey asterisks represent significance of the model comparisons between the two experimental conditions between groups. The bracketed black asterisk represents the paired comparison between all manipulated trials (sad and happy) versus the neutral, non-manipulated (i.e., baseline) condition. * represents p < 0.05, ** p < 0.01, *** p < 0.001. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4 .

 4 Fig. 4. Main results depending on acoustic compensation (i.e. implicit detection) in the group of explicit non-detectors. (A) Normalized valence ratings, (B) normalized confidence ratings, (C) normalized arousal ratings and (D) fundamental frequency of the modified voice (i.e. output of the VFM) depending on condition (happy vs. sad) and implicit detection (i.e., acoustic compensation). Ratings were normalized with respect to neutral trials, so the neutral condition (that is therefore equal to zero) is not shown. Error bars represent confidence intervals. White asterisks represent significance of the model comparisons between experimental conditions (happy and sad VFM) and the baseline condition (neutral), black asterisks represent significance of the model comparisons between the two experimental conditions. * p < 0.05, ** p < 0.01, *** p < 0.001.

Louise Goupil :

 Goupil Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing -original draft, Writing -review & editing. Petter Johansson: Conceptualization, Writing -review & editing, Supervision. Lars Hall Johansson: Conceptualization, Writing -review & editing, Supervision. Jean-Julien Aucouturier: Conceptualization, Methodology, Software, Resources, Writing -review & editing, Supervision, Funding acquisition.

  .,

	Table 1
	Explicit Detection results.
	Detection level	Description	Number of participants
	1	you intentionally manipulated my voice	2
	2	my voice was

higher / lower sometimes, and it changed during the experiment 14 3 my voice sounded strange sometimes, and it changed during the experiment 0 4 my voice sounded strange, and it was not only because I am not used to hearing myself through headphones 1 5 my voice sounded strange, but it is probably because I am not used to hearing myself through headphones 8 6 my voice was absolutely fine 23 2014).

Acknowledgments

The authors thank Louise Vasa and Lou Seropian for their help with collecting the data, as well as Marco Liuni, Pablo Arias, and Laura Rachman for their input regarding the voice transformation algorithm. Experimental data were collected and ethical approval obtained from the INSEAD/ Sorbonne University Center for Behavioural Science. This work was funded by ERC StG CREAM 335536 and ANR 2017 REFLETS (to J.J.A.), a H2020 Marie Skłodowska-Curie actions post-doctoral grant (845859 -JDIL, to L.G.), and a Swedish Research Council grant (2014-137, to PJ and LH).

Appendix A. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.concog.2020.103072.


Student Member, IEEE Jordi Vilà Valls
email: jordi.vilavalls@gipsa-lab.inpg.fr

Jean-Marc Brossier
email: jean-marc.brossier@gipsa-lab.inpg.fr

Laurent Ros
email: laurent.ros@gipsa-lab.inpg.fr

Extended Kalman Filter for Oversampled Dynamical Phase Offset Estimation

Keywords: Phase estimation, Extended Kalman Filter, oversampling, carrier synchronization, GALILEO, BOC

In this paper, we present an application of the Extended Kalman Filter for the on-line estimation of a dynamical carrier phase offset. The novel approach implies deriving the filter in an oversampled scenario in a digital receiver. We consider a Brownian phase evolution in a Data Aided scenario. Our numerical results using a BOC shaping pulse show that using the oversampled signal for estimating the phase offset we can obtain better performances than using a classical synchronizer.

I. INTRODUCTION

Synchronization is a fundamental part in modern digital receivers. A synchronizer has to estimate some parameters, such as carrier frequency, carrier phase and timing epoch. This knowledge is required to recover the signal of interest correctly. In this paper we focus our attention on the phase estimation problem. Many methods for estimating the phase introduced by an unknown channel have been proposed over the past decades, from Phase Locked Loops (PLL) to the most sophisticated signal processing techniques. The Kalman Filter (KF) [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF], [START_REF] Kalman | New results in linear filtering and prediction theory[END_REF], presented in early 1960s is one of the mostly used techniques for parameter estimation in linear gaussian problems. We can find extensive discussion on the KF in [START_REF] Anderson | Optimal Filtering[END_REF] and [START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF]. When dealing with nonlinear filtering problems, the Extended Kalman Filter (EKF) approximates the problem to apply the KF solution. Some contributions show the use of EKF for carrier phase recovery and frequency tracking [START_REF] Kootsookos | An extended kalman filter for demodulation of polynomial phase signals[END_REF]- [START_REF] Cheng | GPS Carrier phase measurement method based on data transition detection in high dynamic circumstance[END_REF]. To our knowledge, the EKF has never been applied to oversampled phase estimation for Binary Offset Carrier (BOC) shaped signals.

In this contribution, we investigate the use of an Extended Kalman Filter for carrier estimation in a Data Aided (DA) scenario. We consider an oversampled signal model after receiver matched filter, this implies having a coloured reception noise. This scenario is standard in radio-localization from a satellite signal. In [START_REF] Vilà | On-line bayesian Cramér-Rao bound for oversampled dynamical phase offset estimation[END_REF], we have shown the potential gain for phase estimation provided by the use of the fractionaly-spaced signal after matched filter, instead of the symbol time-spaced signal. This was done by deriving a closed-form expression of the on-line Bayesian Cramér-Rao Bound (BCRB) for the oversampled dynamical phase estimation. Now, our goal is to propose an EKF based algorithm which can approach this bound. We have thus to jointly estimate the coloured noise and the phase offset because the EKF doesn't take it into account. In Section II, we set the signal model. In Section III, we derive the expressions of the filter in the oversampled phase estimation scenario. In Section IV we recall the BCRB for this estimation problem. Finally in Section V, the numerical results for the EKF resulting from BPSK transmission are presented and interpreted.

N otations: italic indicates a scalar quantity, as in a; boldface indicates a vector quantity, as in a and capital boldface indicates a matrix quantity as in A. The (k, l) th entry of a matrix A is denoted [A] k,l . The matrix transpose and selfadjoint operators are denoted by the superscripts T and H respectively as in A T and A H . ℜ(•), ℑ(•) and (•) * are the real part, the imaginary part and conjugate of a complex number or matrix, respectively. E x denotes the expectation over x.

II. SIGNAL MODEL

We propose the signal model for the transmission of a known sequence {a m } m∈Z over an Additive White Gaussian Noise (AWGN) channel affected by a carrier phase offset θ(t).

A. Oversampled Signal Model 1) Discrete-time general formulation: the received complex baseband signal after matched filtering is

y(t) = T m a m Π(t -mT ) e iθ(t) + n(t) * Π * (-t) (1
) where T, Π(t) and n(t) stands for the symbol period, shaping pulse and circular gaussian noise with a known bilateral power spectral density (psd) N 0 .

We define the filtered noise

b(t) = [n(t)] * Π * (-t) (2)
We also define gm (t) as

gm (t) = T +∞ -∞ Π * (-α)e iθ(t-α+mT ) Π(t -α)dα (3)
Then the received signal can be written as

y(t) = m a m gm (t -mT ) + b(t) (4)
Hereafter we suppose a shaping pulse with support in [0, T ] and a slow varying phase evolution during a period T . In this case we can approximate gm (t) by gm (t) ≈ g(t)e iθ(t+(m+ 1 2 )T ) [START_REF] Anderson | Optimal Filtering[END_REF] where

g(t) = T 0 -T Π * (-α)Π(t -α)dα (6)
If the received signal is fractionally-spaced at t k = k T S + τ , where S is an integer oversampling factor and τ a known offset from the optimum sampling instants (we suppose 0 ≤ τ ≤ T S ), we have that

y " k T S + τ « = X m amgm " k T S + τ -mT « + b " k T S + τ « (7)
and from (eq. 5) we have that

y k T S + τ = e iθ(k T S +τ + T 2 ) A k + b k T S + τ (8)
where

A k = m a m g k T S + τ -mT . (9)
We can finally write the received oversampled signal as

y k = A k e iθ k + b ′ k ( 10
)
where k refers to t k instants. We can define the symbol index p = ⌊ k S ⌋, or equivalently, k = pS + s with s the sample inside the symbol period. We note that s = 0, • • • , S -1.

Note that the noise b ′ k is coloured with variance σ 2 n , where

σ 2 n = N 0 × g(0)
T is the variance of the AWGN noise n(t) measured in the noise equivalent bandwidth of the receiver filter Π * (-t).

2) Discrete-time re-formulation for the noise: the T S -spaced sequence of noise, {b ′ k } k∈Z , is defined in the previous section from an analog noise n(t). Our motivation now is to replace this time serie by another {b k } k∈Z with the same statistical properties, but which can be obtained entirely by a discretetime formulation. This will be useful for the final state-space model formulation. We can write that

b ′ k = T 0 n α + k T S + τ Π * (α) dα = S-1 j=0 (j+1) T S j T S n α + k T S + τ Π * (α) dα
We define Γ as the covariance matrix of the observation noise. If we have N measurements the matrix is N × N and depends on the oversampling factor S.

The random variables

Z k,j = (j+1) T S j T S n α + k T S + τ Π * (α) dα
are zero-mean gaussian distributed. For a fixed k, Z k,j are independent in j. Their variance is equal to

E |Z k,j | 2 = N 0 (j+1) T S j T S |Π (α)| 2 dα
We define a zero-mean, unit variance, gaussian i.i.d sequence n k and

Π j = N 0 (j+1) T S j T S |Π (α)| 2 dα 1 2
Hence, the noise samples b ′ k have the same statistical properties than samples b k obtained by a T S -spaced filtering of the time serie n k :

b k = S-1 j=0 Π j n k-j-1 (11)

B. Phase-offset Evolution Model

In practice we must consider jitters introduced by clocks imperfections. To take it into account we consider a Brownian phase-offset evolution [START_REF] Amblard | Phase tracking: what do we gain from optimality?Particle filtering versus phase-locked loops[END_REF]

θ k = θ k-1 + w k k ≥ 2 (12)
where w k is an i.i.d. zero-mean Gaussian noise with known variance σ 2 w S . Here σ 2 w stands for the variance growth of the phase noise in one symbol period. The N × N covariance matrix of the phase-offset evolution is Σ.

C. State-Space Model

When using an optimal filtering approach a state-space model formulation is needed. Moreover, as we want to take into account that the observation noise is not white, we must include it into the state evolution. We consider a sliding vector

ν k ν k-1 • • • ν k-S+1
T over an i.i.d. noise n k , the evolution of this vector can be written as

2 6 6 4
ν k ν k-1 . . . ν k-S+1 3
7 7 5 = 2 6 6 4 0 • • • • • • 0 1 0 • • • 0 0 . . . . . . 1 0 3 7 7 5 2 6 6 4 ν k-1 ν k-2 . . . ν k-S 3 7 7 5 + 2 6 6 4
n k 0 . . . 0 3 7 7 5 (13)
The coloured noise b k is

b k = [Π 0 • • • Π S-1 ]      ν k-1 ν k-2 . . . ν k-S      (14)
The state to be considered includes the phase-offset and the coloured noise:

x k = ˆθk b k ν k • • • ν k-S+1 ˜T . The state evolution is x k = Mx k-1 + w k ( 15
)
where

M = 2 6 6 6 6 6 6 6 6 4
1 0 0 • • • 0 0 0 Π0 ΠS-1 0 0 0 0 • • • 0 . . . . . . 1 . . . 0 0 • • • 0 1 0 3 7 7 7 7 7 7 7 7 5 (16)
and

w k = w k 0 n k 0 • • • 0 T ( 17
)
The observation equation can be written as

y k = A k exp i 1 0 • • • 0 T x k + 0 1 0 • • • 0 T x k (18)
We note that the state equation is linear and the observation equation depends non-linearly on the state. With this formulation we have no observation noise because we have included it in the state.

III. EXTENDED KALMAN FILTER

In the sequel we introduce the well known EKF. We can find the general EKF expressions in [START_REF] Anderson | Optimal Filtering[END_REF]. Then we derive the expressions on the oversampled carrier phase estimation scenario.

We have a system described by the following state-space equations pair

x k+1 = f k (x k ) + w k y k = g k (x k ) + v k (19)
where x k is the state vector, w k is a zero-mean white noise with covariance matrix Q k , y k is the observation vector at time k which is a partial and noisy observation of the state x k and v k is the observation noise with covariance matrix R k Both noises w k and v k are supposed to be uncorrelated. The functions f k (•) and g k (•) can be non-linear in a general case.

For Gaussian, linear state models, the KF gives the best Mean Square Error (MSE) estimation of the state x k from observations up to time k.

We note x k|m , the estimation of x k from the observations up to time m, x k|m = x kx k|m , the estimation error and P k|m = E x k|m x T k|m , the covariance matrix of the estimation error. The EKF gives us the estimator x k|k in a recursive way. The main idea in the EKF is to linearize the state-space equations at each iteration in order to transform the filtering problem into a usual Kalman one.

So we need to compute

∂f k (x k ) ∂x k ; ∂g k (x k ) ∂x k

A. EKF for Dynamical Phase-Offset Estimation

Here we consider the oversampled phase estimation scenario. So our state-space model is the one presented in Section II (eqs. 15,18).

As the state equation is linear we have directly that

∂f k-1 (x k ) ∂x k = M
The phase noise covariance Q is independent from k and has only two non-zero elements, namely

[Q] 1,1 = σ 2 w S ; [Q] 3,3 = σ 2 n
As we introduced the coloured noise b k into the state, there is no observation noise and the covariance matrix R is null. The observation equation is non-linear with the state, so we have to apply a linearization. We define

g = ∂g k `b x k|k-1 ∂x k = h iA k e i b θ k|k-1 1 0 • • • 0 i T (20)
Replacing these expressions into the general EKF expressions ( [START_REF] Anderson | Optimal Filtering[END_REF]) leads to the oversampled algorithm:

8 > > > > > > > > > > > > < > > > > > > > > > > > > : P k|k-1 = MP k-1|k-1 M H + Q k-1 b x k|k-1 = Mb x k-1|k-1 K k = P k|k-1 g H ˘gP k|k-1 g H ¯-1 P k|k = [I -K k g] P k|k-1 b x k|k = b x k|k-1 + K k h y k -A k e i b θ k|k-1 -b b k|k-1 i (21)
IV. BAYESIAN CRAM ÉR-RAO BOUND

In this section we recall the expression of the on-line Bayesian Cramér-Rao Bound (BCRB) for an oversampled phase estimation problem [START_REF] Vilà | On-line bayesian Cramér-Rao bound for oversampled dynamical phase offset estimation[END_REF]. This bound is particularly suited for problems where an a priori information is available [START_REF] Bay | Analytic and Asymptotic Analysis of Bayesian Cramér-Rao Bound for Dynamical Phase Offset Estimation[END_REF].

The BCRB matrix can be written as

BCRB = {B} -1 = B D + B P -1 (22)
We have to compute two terms. The first one, B D , represents the average information about θ brought by the observations y,

B D k,l = 2ℜ A * l A k Γ -1 k,l e Ψ ( 23
)
where The second term, B P , represents the information available from the prior knowledge on θ,

Ψ = - 1 2 Σ -1 k,k + Σ -1 l,l -2 Σ -1 k,l (24)
B P = 1 σ 2 w /S 0 B B B B B B B @ 1 -1 0 • • • 0 -1 2 -1 . . . . . . 0 . . . . . . . . . 0 . . . -1 2 -1 0 • • • 0 -1 1 1 C C C C C C C A (25)
In the general case, the on-line BCRB associated to obser-

vation vector y = [y 1 • • • y N ] is equal to entry (N, N ) of the BCRB matrix, [BCRB] N,N .
As we analyse the estimation problem in a DA scenario the bound depends on the transmitted sequence a. In this paper we suppose the transmission of a known sequence to analyse the performance of the proposed algorithm and the bound. We note that, contrary to [START_REF] Vilà | On-line bayesian Cramér-Rao bound for oversampled dynamical phase offset estimation[END_REF] where the proposed bound was the minimum over a set of sequences, the BCRB is computed in this paper over the transmitted known sequence.

When S > 1 we have more than one sample/symbol. In this case we have different bounds depending on the position inside the symbol so the BCRB depends on the couple (S, s). If we are interested in the optimal estimation values we set s = 0. So the bound is obtained as

BCRB(S, s) = [BCRB(a)] N,N (26)
with N = (M -1) * S +s+1. M corresponds to the number of transmitted symbols, so the length of the known sequence in our case.

V. DISCUSSION

In this section we show the behaviour of the EKF by considering different scenarios. We assume the transmission over an AWGN channel of a M-sequences of length 511 bits, generated using a Linear Feedback Shift Register (LFSR) with characterisic polynomial [1021] 8 (octal representation). We consider three oversampling factors (S = 1,2 and 4) and a BOC shaping pulse (see figure 1). BOC shaping pulse is used in Galileo positioning system.

In the figures presented we plot the Mean Square Error (MSE) obtained by simulation versus the Signal to Noise Ratio (SNR). The SNR corresponds to the Carrier to Noise Ratio ( C N ) at the input of the receiver. In our case, as shaping pulse and symbols a k are normalised (i.e σ 2 a = 1; g(0) = 1) this ratio is simply

C N = 1 σ 2 n
. We compute the MSE for the Tspaced optimal estimation values corresponding to the s = 0 case for all oversampling factors S. We also give the on-line BCRB for each case as a reference.

Figure 2 and 3 superimpose versus the SNR, the on-line BCRB (see eq.( 26)) and the EKF. For figure 2, we have a slow varying phase with variance σ 2 w = 0.001 rad 2 and for figure 3, we have a phase with a fast evolution, σ 2 w = 0.01 rad 2 . In both scenarios there's no offset from the optimal sampling instants, τ = 0.

One can see that for S = 1 the performance of the EKF is the same as the theoretical result of the BCRB. For S = 2 the performance of the algorithm are slightly looser when comparing to the bound. For S = 4 we obtain the same or slightly better performance as in the S = 2 case. In this case if we compare the algorithm with the bound we can see that the performance poorer. The gain increases with the oversampling factor S and the interest of oversampling becomes clear at low SNR. The gain due to oversampling decreases as the SNR increases.

In figure 4 we analyse the EKF behaviour for a fixed SNR versus phase-noise variance. We present a scenario with a low SNR value, SNR = 0dB. Here we can still measure the gain given by the oversampling and the good performance of the algorithm. The gain obtained with the oversampling is greater at weak σ 2 w . We also note that the performance of the algorithm at weak phase noise variance is really close from the bound. At very high σ 2 w the performances become poorer compared to the bound. This is probably because for high SNR, the modeling error in the EKF linear approximation (see eq. 20) is not neglictible with respect to the noise level.

Figure 5 superimposes versus the SNR, the on-line BCRB and the EKF for a slow varying phase evolution scenario, σ 2 w = 0.001, and a non-null offset τ = T 8 for S = 1 and S = 2. We show in the same figure the performance of the EKF for a null offset τ = 0 as a reference. We can see that the performances for both the bound and the algorithm are looser when having a non-null offset τ . In general the performances decrease when increasing τ (0 ≤ τ ≤ T S ). We also note that the gain between different oversampling factors is greater at high SNR when having a non-null offset.

VI. CONCLUSION

In this contribution, we have presented the Extended Kalman Filter for a realistic dynamical carrier phase estimation in an oversampled scenario. We have presented numerical results using a time limited pulse as done in satellite positioning systems. In such scenario, where the Shannon sampling theorem is not respected, we have shown the interest of using a fractionally-spaced method for phase estimation. The interest of using this algorithm with the oversampled signal becomes clear at low SNR. The results obtained with the EKF are close to the theoretical bound for slow and moderate varying phase evolutions.

Fig. 1 .

 1 Fig. 1. BOC shaping function Π(t) and its autocorrelation g(t)

Fig. 2 .

 2 Fig. 2. EKF and BCRB versus the SNR for three different oversampling factors S = 1, 2 and 4, with a phase-noise variance σ 2 w = 0.001 rad 2 .

Fig. 3 .Fig. 4 .

 34 Fig. 3. EKF and BCRB versus the SNR for three different oversampling factors S = 1, 2 and 4, with a phase-noise variance σ 2 w = 0.01 rad 2 .

Fig. 5 .

 5 Fig. 5. EKF and BCRB in presence of a non nul offset for the sampling instants for two different oversampling factors S = 1 and 2, SNR = 0dB.

ACKNOWLEDGEMENT

This work was supported in part by the French ANR (Agence Nationale de la Recherche), LURGA project.