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Fast Fourier transform

A Fast Fourier Transform (FFT) is an algorithm that computes the Discrete Fourier Transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from $O(n^2)$, which arises if one simply applies the definition of DFT, to $O(n \log n)$ , where $n$ is the data size. The difference in speed can be enormous, especially for long data sets where $n$ may be in the thousands or millions. In the presence of round-off error, many FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.

Fast Fourier transforms are widely used for applications in engineering, music, science, and mathematics. The basic ideas were popularized in 1965, but some algorithms had been derived as early as 1805. In 1994, Gilbert Strang described the FFT as "the most important numerical algorithm of our lifetime", and it was included in Top 10 Algorithms of 20th Century by the IEEE magazine Computing in Science & Engineering.

The best-known FFT algorithms depend upon the factorization of $n$, but there are FFTs with $$O(n \log n)$$ complexity for all, even prime, $n$. Many FFT algorithms depend only on the fact that $e^{-2\pi i/n}$ is an $n$'th primitive root of unity, and thus can be applied to analogous transforms over any finite field, such as number-theoretic transforms. Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a $1/n$ factor, any FFT algorithm can easily be adapted for it.

History
The development of fast algorithms for DFT can be traced to Carl Friedrich Gauss's unpublished 1805 work on the orbits of asteroids Pallas and Juno. Gauss wanted to interpolate the orbits from sample observations; his method was very similar to the one that would be published in 1965 by James Cooley and John Tukey, who are generally credited for the invention of the modern generic FFT algorithm. While Gauss's work predated even Joseph Fourier's 1822 results, he did not analyze the method's complexity, and eventually used other methods to achieve the same end.

Between 1805 and 1965, some versions of FFT were published by other authors. Frank Yates in 1932 published his version called interaction algorithm, which provided efficient computation of Hadamard and Walsh transforms. Yates' algorithm is still used in the field of statistical design and analysis of experiments. In 1942, G. C. Danielson and Cornelius Lanczos published their version to compute DFT for x-ray crystallography, a field where calculation of Fourier transforms presented a formidable bottleneck. While many methods in the past had focused on reducing the constant factor for $O(n^2)$ computation by taking advantage of "symmetries", Danielson and Lanczos realized that one could use the "periodicity" and apply a "doubling trick" to "double [$n$] with only slightly more than double the labor", though like Gauss they did not do the analysis to discover that this led to $O(n \log n)$  scaling.

James Cooley and John Tukey independently rediscovered these earlier algorithms and published a more general FFT in 1965 that is applicable when $n$ is composite and not necessarily a power of 2, as well as analyzing the $O(n \log n)$ scaling. Tukey came up with the idea during a meeting of President Kennedy's Science Advisory Committee where a discussion topic involved detecting nuclear tests by the Soviet Union by setting up sensors to surround the country from outside. To analyze the output of these sensors, an FFT algorithm would be needed. In discussion with Tukey, Richard Garwin recognized the general applicability of the algorithm not just to national security problems, but also to a wide range of problems including one of immediate interest to him, determining the periodicities of the spin orientations in a 3-D crystal of Helium-3. Garwin gave Tukey's idea to Cooley (both worked at IBM's Watson labs) for implementation. Cooley and Tukey published the paper in a relatively short time of six months. As Tukey did not work at IBM, the patentability of the idea was doubted and the algorithm went into the public domain, which, through the computing revolution of the next decade, made FFT one of the indispensable algorithms in digital signal processing.

Definition
Let $$x_0, \ldots, x_{n-1}$$ be complex numbers. The DFT is defined by the formula

 * $$ X_k = \sum_{m=0}^{n-1} x_m e^{-i2\pi k m/n} \qquad k = 0,\ldots,n-1, $$

where $$e^{i 2\pi/n}$$ is a primitive $n$'th root of 1.

Evaluating this definition directly requires $O(n^2)$ operations: there are $n$ outputs $Xk$, and each output requires a sum of $n$ terms. An FFT is any method to compute the same results in $O(n \log n)$ operations. All known FFT algorithms require $O(n \log n)$ operations, although there is no known proof that lower complexity is impossible.

To illustrate the savings of an FFT, consider the count of complex multiplications and additions for $n=4096$ data points. Evaluating the DFT's sums directly involves $n^2$ complex multiplications and $n(n-1)$  complex additions, of which $O(n)$  operations can be saved by eliminating trivial operations such as multiplications by 1, leaving about 30 million operations. In contrast, the radix-2 Cooley–Tukey algorithm, for $n$ a power of 2, can compute the same result with only $(n/2)\log_2(n)$ complex multiplications (again, ignoring simplifications of multiplications by 1 and similar) and $n\log_2(n)$  complex additions, in total about 30,000 operations — a thousand times less than with direct evaluation. In practice, actual performance on modern computers is usually dominated by factors other than the speed of arithmetic operations and the analysis is a complicated subject (for example, see Frigo & Johnson, 2005), but the overall improvement from $O(n^2)$ to $O(n \log n)$  remains.

Cooley–Tukey algorithm
By far the most commonly used FFT is the Cooley–Tukey algorithm. This is a divide-and-conquer algorithm that recursively breaks down a DFT of any composite size $n = n_1n_2$ into $n_1$  smaller DFTs of size $n_2$, along with $$O(n)$$ multiplications by complex roots of unity traditionally called twiddle factors (after Gentleman and Sande, 1966).

This method (and the general idea of an FFT) was popularized by a publication of Cooley and Tukey in 1965, but it was later discovered that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited forms).

The best known use of the Cooley–Tukey algorithm is to divide the transform into two pieces of size $n/2$ at each step, and is therefore limited to power-of-two sizes, but any factorization can be used in general (as was known to both Gauss and Cooley/Tukey ). These are called the radix-2 and mixed-radix cases, respectively (and other variants such as the split-radix FFT have their own names as well). Although the basic idea is recursive, most traditional implementations rearrange the algorithm to avoid explicit recursion. Also, because the Cooley–Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT, such as those described below.

Other FFT algorithms
There are FFT algorithms other than Cooley–Tukey.

For $n = n_1n_2$ with coprime $n_1$  and $n_2$, one can use the prime-factor (Good–Thomas) algorithm (PFA), based on the Chinese remainder theorem, to factorize the DFT similarly to Cooley–Tukey but without the twiddle factors. The Rader–Brenner algorithm (1976) is a Cooley–Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost of increased additions and reduced numerical stability; it was later superseded by the split-radix variant of Cooley–Tukey (which achieves the same multiplication count but with fewer additions and without sacrificing accuracy). Algorithms that recursively factorize the DFT into smaller operations other than DFTs include the Bruun and QFT algorithms. (The Rader–Brenner and QFT algorithms were proposed for power-of-two sizes, but it is possible that they could be adapted to general composite $n$. Bruun's algorithm applies to arbitrary even composite sizes.) Bruun's algorithm, in particular, is based on interpreting the FFT as a recursive factorization of the polynomial $$z^n-1$$, here into real-coefficient polynomials of the form $$z^m-1$$ and $$z^{2m} + az^m + 1$$.

Another polynomial viewpoint is exploited by the Winograd FFT algorithm, which factorizes $$z^n-1$$ into cyclotomic polynomials—these often have coefficients of 1, 0, or −1, and therefore require few (if any) multiplications, so Winograd can be used to obtain minimal-multiplication FFTs and is often used to find efficient algorithms for small factors. Indeed, Winograd showed that the DFT can be computed with only $$O(n)$$ irrational multiplications, leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes; this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardware multipliers. In particular, Winograd also makes use of the PFA as well as an algorithm by Rader for FFTs of prime sizes.

Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime $n$, expresses a DFT of prime size $n$ as a cyclic convolution of (composite) size $n – 1$, which can then be computed by a pair of ordinary FFTs via the convolution theorem (although Winograd uses other convolution methods). Another prime-size FFT is due to L. I. Bluestein, and is sometimes called the chirp-z algorithm; it also re-expresses a DFT as a convolution, but this time of the same size (which can be zero-padded to a power of two and evaluated by radix-2 Cooley–Tukey FFTs, for example), via the identity

 * $$nk = -\frac{(k-n)^2} 2 + \frac{n^2} 2 + \frac{k^2} 2.$$

Hexagonal fast Fourier transform (HFFT) aims at computing an efficient FFT for the hexagonally-sampled data by using a new addressing scheme for hexagonal grids, called Array Set Addressing (ASA).

FFT algorithms specialized for real or symmetric data
In many applications, the input data for the DFT are purely real, in which case the outputs satisfy the symmetry

 * $$X_{n-k} = X_k^*$$

and efficient FFT algorithms have been designed for this situation (see e.g. Sorensen, 1987). One approach consists of taking an ordinary algorithm (e.g. Cooley–Tukey) and removing the redundant parts of the computation, saving roughly a factor of two in time and memory. Alternatively, it is possible to express an even-length real-input DFT as a complex DFT of half the length (whose real and imaginary parts are the even/odd elements of the original real data), followed by $$O(n)$$ post-processing operations.

It was once believed that real-input DFTs could be more efficiently computed by means of the discrete Hartley transform (DHT), but it was subsequently argued that a specialized real-input DFT algorithm (FFT) can typically be found that requires fewer operations than the corresponding DHT algorithm (FHT) for the same number of inputs. Bruun's algorithm (above) is another method that was initially proposed to take advantage of real inputs, but it has not proved popular.

There are further FFT specializations for the cases of real data that have even/odd symmetry, in which case one can gain another factor of roughly two in time and memory and the DFT becomes the discrete cosine/sine transform(s) (DCT/DST). Instead of directly modifying an FFT algorithm for these cases, DCTs/DSTs can also be computed via FFTs of real data combined with $$O(n)$$ pre- and post-processing.

Bounds on complexity and operation counts
A fundamental question of longstanding theoretical interest is to prove lower bounds on the complexity and exact operation counts of fast Fourier transforms, and many open problems remain. It is not rigorously proved whether DFTs truly require $\Omega(n \log n)$ (i.e., order $$n \log n$$ or greater) operations, even for the simple case of power of two sizes, although no algorithms with lower complexity are known. In particular, the count of arithmetic operations is usually the focus of such questions, although actual performance on modern-day computers is determined by many other factors such as cache or CPU pipeline optimization.

Following work by Shmuel Winograd (1978), a tight $$\Theta(n)$$ lower bound is known for the number of real multiplications required by an FFT. It can be shown that only $4n - 2\log_2^2(n) - 2\log_2(n) - 4$ irrational real multiplications are required to compute a DFT of power-of-two length $$n = 2^m$$. Moreover, explicit algorithms that achieve this count are known (Heideman & Burrus, 1986; Duhamel, 1990 ). However, these algorithms require too many additions to be practical, at least on modern computers with hardware multipliers (Duhamel, 1990; Frigo & Johnson, 2005).

A tight lower bound is not known on the number of required additions, although lower bounds have been proved under some restrictive assumptions on the algorithms. In 1973, Morgenstern proved an $$\Omega(n \log n)$$ lower bound on the addition count for algorithms where the multiplicative constants have bounded magnitudes (which is true for most but not all FFT algorithms). Pan (1986) proved an $$\Omega(n \log n)$$ lower bound assuming a bound on a measure of the FFT algorithm's "asynchronicity", but the generality of this assumption is unclear. For the case of power-of-two $n$, Papadimitriou (1979) argued that the number $n \log_2 n$ of complex-number additions achieved by Cooley–Tukey algorithms is optimal under certain assumptions on the graph of the algorithm (his assumptions imply, among other things, that no additive identities in the roots of unity are exploited). (This argument would imply that at least $2N \log_2 N$ real additions are required, although this is not a tight bound because extra additions are required as part of complex-number multiplications.) Thus far, no published FFT algorithm has achieved fewer than $n \log_2 n$  complex-number additions (or their equivalent) for power-of-two $n$.

A third problem is to minimize the total number of real multiplications and additions, sometimes called the "arithmetic complexity" (although in this context it is the exact count and not the asymptotic complexity that is being considered). Again, no tight lower bound has been proven. Since 1968, however, the lowest published count for power-of-two $n$ was long achieved by the split-radix FFT algorithm, which requires $4n\log_2(n) - 6n + 8$ real multiplications and additions for $n > 1$. This was recently reduced to $\sim \frac{34}{9} n \log_2 n$ (Johnson and Frigo, 2007; Lundy and Van Buskirk, 2007 ). A slightly larger count (but still better than split radix for $n ≥ 256$) was shown to be provably optimal for $n ≤ 512$ under additional restrictions on the possible algorithms (split-radix-like flowgraphs with unit-modulus multiplicative factors), by reduction to a satisfiability modulo theories problem solvable by brute force (Haynal & Haynal, 2011).

Most of the attempts to lower or prove the complexity of FFT algorithms have focused on the ordinary complex-data case, because it is the simplest. However, complex-data FFTs are so closely related to algorithms for related problems such as real-data FFTs, discrete cosine transforms, discrete Hartley transforms, and so on, that any improvement in one of these would immediately lead to improvements in the others (Duhamel & Vetterli, 1990).

Approximations
All of the FFT algorithms discussed above compute the DFT exactly (i.e. neglecting floating-point errors). A few "FFT" algorithms have been proposed, however, that compute the DFT approximately, with an error that can be made arbitrarily small at the expense of increased computations. Such algorithms trade the approximation error for increased speed or other properties. For example, an approximate FFT algorithm by Edelman et al. (1999) achieves lower communication requirements for parallel computing with the help of a fast multipole method. A wavelet-based approximate FFT by Guo and Burrus (1996) takes sparse inputs/outputs (time/frequency localization) into account more efficiently than is possible with an exact FFT. Another algorithm for approximate computation of a subset of the DFT outputs is due to Shentov et al. (1995). The Edelman algorithm works equally well for sparse and non-sparse data, since it is based on the compressibility (rank deficiency) of the Fourier matrix itself rather than the compressibility (sparsity) of the data. Conversely, if the data are sparse—that is, if only $k$ out of $n$ Fourier coefficients are nonzero—then the complexity can be reduced to $$O(k \log n \log n/k)$$, and this has been demonstrated to lead to practical speedups compared to an ordinary FFT for $n/k > 32$ in a large-$n$ example ($n = 222$) using a probabilistic approximate algorithm (which estimates the largest $k$ coefficients to several decimal places).

Accuracy
FFT algorithms have errors when finite-precision floating-point arithmetic is used, but these errors are typically quite small; most FFT algorithms, e.g. Cooley–Tukey, have excellent numerical properties as a consequence of the pairwise summation structure of the algorithms. The upper bound on the relative error for the Cooley–Tukey algorithm is $O(\varepsilon \log n)$, compared to $O(\varepsilon n^{3/2})$ for the naïve DFT formula, where $epsilon$ is the machine floating-point relative precision. In fact, the root mean square (rms) errors are much better than these upper bounds, being only $O(\varepsilon \sqrt{\log n})$ for Cooley–Tukey and $O(\varepsilon \sqrt{n})$  for the naïve DFT (Schatzman, 1996). These results, however, are very sensitive to the accuracy of the twiddle factors used in the FFT (i.e. the trigonometric function values), and it is not unusual for incautious FFT implementations to have much worse accuracy, e.g. if they use inaccurate trigonometric recurrence formulas. Some FFTs other than Cooley–Tukey, such as the Rader–Brenner algorithm, are intrinsically less stable.

In fixed-point arithmetic, the finite-precision errors accumulated by FFT algorithms are worse, with rms errors growing as $O(\sqrt{n})$ for the Cooley–Tukey algorithm (Welch, 1969). Achieving this accuracy requires careful attention to scaling to minimize loss of precision, and fixed-point FFT algorithms involve rescaling at each intermediate stage of decompositions like Cooley–Tukey.

To verify the correctness of an FFT implementation, rigorous guarantees can be obtained in $O(n \log n)$ time by a simple procedure checking the linearity, impulse-response, and time-shift properties of the transform on random inputs (Ergün, 1995).

The values for intermediate frequencies may be obtained by various averaging methods.

Multidimensional FFTs
As defined in the multidimensional DFT article, the multidimensional DFT

 * $$X_\mathbf{k} = \sum_{\mathbf{n}=0}^{\mathbf{N}-1} e^{-2\pi i \mathbf{k} \cdot (\mathbf{n} / \mathbf{N})} x_\mathbf{n}$$

transforms an array $xn$ with a $d$-dimensional vector of indices $\mathbf{n} = \left(n_1, \ldots, n_d\right)$ by a set of $d$ nested summations (over $n_j = 0 \ldots N_j - 1$  for each $j$), where the division $\mathbf{n} / \mathbf{N} = \left(n_1/N_1, \ldots, n_d/N_d\right)$  is performed element-wise. Equivalently, it is the composition of a sequence of d sets of one-dimensional DFTs, performed along one dimension at a time (in any order).

This compositional viewpoint immediately provides the simplest and most common multidimensional DFT algorithm, known as the row-column algorithm (after the two-dimensional case, below). That is, one simply performs a sequence of $d$ one-dimensional FFTs (by any of the above algorithms): first you transform along the $n1$ dimension, then along the $n2$ dimension, and so on (actually, any ordering works). This method is easily shown to have the usual $O(n \log n)$ complexity, where $n = n_1 \cdot n_2 \cdots n_d$  is the total number of data points transformed. In particular, there are $n/n1$ transforms of size $n1$, etc., so the complexity of the sequence of FFTs is:

 * $$\begin{align}

& \frac{n}{n_1} O(n_1 \log n_1) + \cdots + \frac{n}{n_d} O(n_d \log n_d) \\[6pt] ={} & O\left(n \left[\log n_1 + \cdots + \log n_d\right]\right) = O(n \log n). \end{align}$$

In two dimensions, the xk can be viewed as an $$n_1 \times n_2$$ matrix, and this algorithm corresponds to first performing the FFT of all the rows (resp. columns), grouping the resulting transformed rows (resp. columns) together as another $$n_1 \times n_2$$ matrix, and then performing the FFT on each of the columns (resp. rows) of this second matrix, and similarly grouping the results into the final result matrix.

In more than two dimensions, it is often advantageous for cache locality to group the dimensions recursively. For example, a three-dimensional FFT might first perform two-dimensional FFTs of each planar "slice" for each fixed n1, and then perform the one-dimensional FFTs along the n1 direction. More generally, an asymptotically optimal cache-oblivious algorithm consists of recursively dividing the dimensions into two groups $(n_1, \ldots, n_{d/2})$ and $(n_{d/2+1}, \ldots, n_d)$  that are transformed recursively (rounding if $d$ is not even) (see Frigo and Johnson, 2005). Still, this remains a straightforward variation of the row-column algorithm that ultimately requires only a one-dimensional FFT algorithm as the base case, and still has $$O(n \log n)$$ complexity. Yet another variation is to perform matrix transpositions in between transforming subsequent dimensions, so that the transforms operate on contiguous data; this is especially important for out-of-core and distributed memory situations where accessing non-contiguous data is extremely time-consuming.

There are other multidimensional FFT algorithms that are distinct from the row-column algorithm, although all of them have $O(n \log n)$ complexity. Perhaps the simplest non-row-column FFT is the vector-radix FFT algorithm, which is a generalization of the ordinary Cooley–Tukey algorithm where one divides the transform dimensions by a vector $\mathbf{r} = \left(r_1, r_2, \ldots, r_d\right)$ of radices at each step. (This may also have cache benefits.) The simplest case of vector-radix is where all of the radices are equal (e.g. vector-radix-2 divides all of the dimensions by two), but this is not necessary. Vector radix with only a single non-unit radix at a time, i.e. $\mathbf{r} = \left(1, \ldots, 1, r, 1, \ldots, 1\right)$, is essentially a row-column algorithm. Other, more complicated, methods include polynomial transform algorithms due to Nussbaumer (1977), which view the transform in terms of convolutions and polynomial products. See Duhamel and Vetterli (1990) for more information and references.

Other generalizations
An $O(n^{5/2} \log n)$ generalization to spherical harmonics on the sphere $S2$ with $n2$ nodes was described by Mohlenkamp, along with an algorithm conjectured (but not proven) to have $O(n^2 \log^2(n))$  complexity; Mohlenkamp also provides an implementation in the libftsh library. A spherical-harmonic algorithm with $O(n^2 \log n)$ complexity is described by Rokhlin and Tygert.

The fast folding algorithm is analogous to the FFT, except that it operates on a series of binned waveforms rather than a series of real or complex scalar values. Rotation (which in the FFT is multiplication by a complex phasor) is a circular shift of the component waveform.

Various groups have also published "FFT" algorithms for non-equispaced data, as reviewed in Potts et al. (2001). Such algorithms do not strictly compute the DFT (which is only defined for equispaced data), but rather some approximation thereof (a non-uniform discrete Fourier transform, or NDFT, which itself is often computed only approximately). More generally there are various other methods of spectral estimation.

Applications
The FFT is used in digital recording, sampling, additive synthesis and pitch correction software.

The FFT's importance derives from the fact that it has made working in the frequency domain equally computationally feasible as working in the temporal or spatial domain. Some of the important applications of the FFT include:

 * fast large-integer multiplication algorithms and polynomial multiplication,
 * efficient matrix–vector multiplication for Toeplitz, circulant and other structured matrices,
 * filtering algorithms (see overlap–add and overlap–save methods),
 * fast algorithms for discrete cosine or sine transforms (e.g. fast DCT used for JPEG and MPEG/MP3 encoding and decoding),
 * fast Chebyshev approximation,
 * solving difference equations,
 * computation of isotopic distributions.
 * modulation and demodulation of complex data symbols using orthogonal frequency division multiplexing (OFDM) for 5G, LTE, Wi-Fi, DSL, and other modern communication systems.

An original application of the FFT in finance particularly in the Valuation of options was developed by Marcello Minenna.

Research areas

 * Big FFTs: With the explosion of big data in fields such as astronomy, the need for 512K FFTs has arisen for certain interferometry calculations. The data collected by projects such as WMAP and LIGO require FFTs of tens of billions of points. As this size does not fit into main memory, so called out-of-core FFTs are an active area of research.
 * Approximate FFTs: For applications such as MRI, it is necessary to compute DFTs for nonuniformly spaced grid points and/or frequencies. Multipole based approaches can compute approximate quantities with factor of runtime increase.
 * Group FFTs: The FFT may also be explained and interpreted using group representation theory allowing for further generalization. A function on any compact group, including non-cyclic, has an expansion in terms of a basis of irreducible matrix elements. It remains active area of research to find efficient algorithm for performing this change of basis. Applications including efficient spherical harmonic expansion, analyzing certain Markov processes, robotics etc.
 * Quantum FFTs: Shor's fast algorithm for integer factorization on a quantum computer has a subroutine to compute DFT of a binary vector. This is implemented as sequence of 1- or 2-bit quantum gates now known as quantum FFT, which is effectively the Cooley–Tukey FFT realized as a particular factorization of the Fourier matrix. Extension to these ideas is currently being explored.

title: Fast Fourier transform


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Fort William, Scotland

Fort William is a town in Lochaber in the Scottish Highlands, located on the eastern shore of Loch Linnhe. At the 2011 census, Fort William had a population of 10,459, making it the second-largest settlement both in the Highland council area and in the whole of the Scottish Highlands; only the city of Inverness has a larger population.

Fort William is a major tourist centre on the Road to the Isles, with Glen Coe just to the south, Ben Nevis and Aonach Mòr to the east, and Glenfinnan to the west. It is a centre for hillwalking and climbing due to its proximity to Ben Nevis and many other Munros. It is also known for its nearby downhill mountain bike track. It is the start/end of both the West Highland Way (Milngavie – Fort William) and the Great Glen Way (a walk/cycleway, Fort William – Inverness).

Around 726 people (7.33% of the population) can speak Gaelic.

Origins
The earliest recorded settlement on the site is a Cromwellian wooden fort built in 1654 as a base for the New Model Army to "pacify" Clan Cameron after the Wars of the Three Kingdoms. The post-1688 revolution fort was named Fort William after William of Orange, who ordered that it be built to control some of the Scottish clans. The settlement that grew around it was called Maryburgh, after his wife Mary II of England. This settlement was later renamed Gordonsburgh, and then Duncansburgh before being renamed Fort William, this time after Prince William, Duke of Cumberland. There have been various suggestions over the years to rename the town (for example, to Invernevis).

The origin of the Gaelic name for Fort William, An Gearasdan, is not recorded but could be a loanword from the English garrison, having entered common usage some time after the royal garrison was established, during the reign of William of Orange or perhaps after the earlier Cromwellian fort.

History
Historically, the Fort William area of Lochaber was old Clan Chattan and then Clan Cameron country, and there were a number of mainly Cameron settlements in the area (such as Blarmacfoldach). Before the building of the fort, Inverlochy was the main local settlement and was also the site of two battles—the first Battle of Inverlochy in 1431 and the second Battle of Inverlochy in 1645.

The town grew in size as a settlement when the fort was constructed to control the population after Oliver Cromwell's invasion during the Wars of the Three Kingdoms, and then to suppress the Jacobite risings after the 1688 overthrow of the House of Stuart.

During the 1745 Jacobite Rising, known as the Forty-Five, Fort William was besieged for two weeks by the Jacobite forces, from 20 March to 3 April 1746. However, although the Jacobites had captured both of the other forts in the chain of three Great Glen fortifications (Fort Augustus and the original Fort George), they failed to take Fort William.

In 1934, the Laggan Dam on the River Spean was completed as part of the Lochaber hydroelectric scheme by Balfour Beatty for the British Aluminium Company. The supervising engineers were the firm of C. S. Meik and William Halcrow, now known as the Halcrow Group. The dam was built to power the aluminium smelter.

During the Second World War, Fort William was the home of HMS St Christopher (shore establishment), which was a training base for Royal Navy Coastal Forces.

Liberty House Group
In November 2016, Sanjeev Gupta's Liberty House Group purchased the Lochaber aluminum plant from the Rio Tinto Group.

On 3 April 2021, it came to light during Gupta's financial troubles involving Greensill Capital that the Jahama Highland Estates (formerly the "Alcan Estate") had been purchased in 2016 as part of the Rio Tinto Mines deal for the Lochaber aluminium plant, because the furnace requires so much power that the smelter is located near a hydroelectric plant, which drains the basin of the 114,000 acre Estate.

The Estate includes the north face of Ben Nevis. According to reports, the Scottish National Party mandated that the Estate never be split from the hydro plant and aluminium smelter, but Gupta ignored them and placed ownership of the Estate in a company that is domiciled on the Isle of Man. The 2016 deal was worth £330 million and was guaranteed by the UK Chancellor of the Exchequer. Conservative finance spokesperson Murdo Fraser was critical of the alleged breach of the SNP agreement and urged the SNP to "take whatever steps are necessary to protect public funds".

Proposed development
A "Waterfront" development was proposed by the council, but failed due to lack of public support. The development would have included a hotel, shops, and housing. The proposed development was slated to take 7 years by the local council, but opponents of the project said that it was unlikely to be completed before 2020. It was announced in April 2010 that the project had been abandoned.

Geography
Originally based on the still-extant village of Inverlochy, the town lies at the southern end of the Great Glen. Fort William lies near the head of Loch Linnhe, one of Scotland's longest sea lochs, beside the mouth of the rivers Nevis and Lochy. They join in the intertidal zone and briefly become one river before discharging to the sea. The town and its suburbs are surrounded by mountains. Its suburbs of Lochyside, Caol and Corpach are on the shore of Loch Eil. It is close to Ben Nevis, the highest mountain in the British Isles, Glen Nevis, and the settlement of Achnaphubuil, which is on the opposite shore of the loch. The original railway station, which was opened on 7 August 1894, was at the south end of the town. The consequence of this decision was that the town was separated from the lochside by railway tracks until the 1970s, when the present by-pass was built, and the station was relocated to the north end. The town is centred on the High Street, which was pedestrianised in the 1990s. Off this, there are several squares: Monzie Square (named after the Cameron Campbells of Monzie, Perthshire, former landowners in the town); Station Square, where the long-since demolished railway station used to be; Gordon Square (named after the Gordons, who in the late 18th century owned land where the town now stands, when the town was named Gordonsburgh); and Cameron Square—formerly known as Town Hall Square. There is also Fraser Square, which is not so square-like, since it now opens out into Middle Street, but which still houses the Imperial Hotel.

The main residential areas of the town are unseen from the High Street and the A82 main road. Upper Achintore and the Plantation spread steeply uphill from above the high street. Inverlochy, Claggan, An-Aird, Lochyside, Caol, Banavie and Corpach outwith (i.e. outside) the town are the other main residential areas. These areas are built on much flatter land than the town.

Fort William is the northern end of the West Highland Way, a long-distance route which runs 95 mi through the Scottish Highlands to Milngavie, on the outskirts of Glasgow, and the start/end point of the Great Glen Way, which runs between Fort William and Inverness.

Glenfinnan, 17 mi away, is home of the Glenfinnan Monument (Jacobite era) and the famous Glenfinnan Viaduct (as seen on a Bank Of Scotland £10 note ). The viaduct has become known to millions in recent years as the "Harry Potter Bridge" after it featured in the films of the books by J.K. Rowling, specifically Harry Potter and the Chamber of Secrets. Glenfinnan has also been used in Charlotte Gray and Highlander.

Just outside the town is a large aluminium plant once operated by Alcan and powered by the Lochaber hydroelectric scheme, in its day the biggest tunnelling project in the world. This was formerly served by the Lochaber Narrow Gauge Railway better known locally as the Puggy Line.

Climate
Fort William has an oceanic climate (Cfb) with moderate, but generally cool, temperatures and abundant precipitation. Fort William is one of the wettest locations in the British Isles, with December being the wettest month.

Transport
The West Highland Line passes through Fort William. Owing to the difficult terrain in the area, the line from Glasgow to the south, enters from the north-east. Trains from Glasgow to Mallaig, the terminus of the line, have to reverse at Fort William railway station.

An overnight train, the Caledonian sleeper, has its terminus at Fort William; this service is known colloquially as "The Deerstalker". The stands for local buses and express coaches are on MacFarlane Way adjacent to the railway station.

The Caledonian Canal connects the Scottish east coast at Inverness with the west coast at Corpach near Fort William.

The Corran Ferry crosses Loch Linnhe, connecting the A82 to A861.

Fort William is located on the A82. The closest motorway access is the junction after Erskine bridge (just off the A82) of the M898 near Erskine Renfrewshire, 98 miles to the south west, joining onto the M8 motorway and the south.

Mountain biking
Just outside the town, parallel to the Nevis Range Gondola, there is a large downhill mountain bike track which attracts thousands of visitors every year, including international competitors and fans.

Each year since 2002, Fort William has hosted a round of UCI Mountain Bike World Cup and, in 2007, it hosted the UCI Mountain Bike & Trials World Championships ('The Worlds'). A trials competition is held at the various courses at the bottom.

Fort William has hosted the World Endurance Mountain Bike Organisation (WEMBO) solo 24-hour championship in 2014 and again in 2018. The event consists of riders racing for a full 24 hours and is won by the rider who has completed the greater number of laps.

Motorcycle trials
Fort William is the home of the Scottish Six Day Motorcycle Trial (SSDT); it is held annually in the first full week of May. It attracts many competitors from all across the globe and, in 2011, the event celebrated its centenary year.

Scottish Six Days Trial

Others
Fort William has two major shinty teams: Fort William Shinty Club and Kilmallie Shinty Club. It also has a football team, Fort William F.C., that competes in the Scottish North Caledonian League and plays home games at Claggan Park. There is also a cricket club at Fort William that participates in the North of Scotland Cricket Association league (NoSCA).

In addition, the town is home to Lochaber Rugby Club and to the Lochaber Yacht Club, a Community Amateur Sports Club that was founded in 1954. The town also has one golf club, called Fort William Golf Club, which has eighteen holes and is open year-round; it also hosts weekly competitions.

As a film location
Movies filmed in or near Fort William include Being Human, Braveheart, Highlander, Restless Natives, Harry Potter and Rob Roy. The TV series Rockface was filmed mainly around Fort William and some scenes of Monarch of the Glen were filmed around Lochaber, although mostly near Newtonmore. Local Hero shot the internal Houston scenes in Fort William.

Festivals
In a celebration of mountains and the culture that surrounds them, and in recognition of the importance of climbing and walking tourism to the town, the Fort William Mountain Festival is held there each year. For a number of years, this volunteer-led festival has concentrated mostly around film but, starting in the Year of Highland Culture – Highland 2007, its scope was widened, and it dropped the word 'film' from its title.

Education
Lochaber High School is the local high school and serves a large catchment area which includes the surrounding villages. West Highland College is part of the University of the Highlands & Islands. It hosts the School of Adventure Studies (SOAS) offering both FE courses & HE honours degrees.

Notable people

 * Danny Alexander (born 1972), the former Chief Secretary to the Treasury and Liberal Democrat MP for Inverness, Nairn, Badenoch and Strathspey. Brought up in Invergarry, a small village near Fort William.
 * Colonel Hugh Cochrane (1829–1884), recipient of the Victoria Cross for actions during the Indian Rebellion of 1857
 * Barry Hutchison, author
 * Charles Kennedy (1959–2015), former leader of the Liberal Democrat party and former Liberal Democrat Member of Parliament (MP) for Ross, Skye and Lochaber. Although born in Inverness, he was brought up and educated in Fort William.
 * Allan MacDonald (Maighstir Alein, An t-Athair Ailean Dòmhnallach) (1859-1905), Roman Catholic priest, folklore collector, pastor in South Uist & Eriskay and poet who helped create modern Scottish Gaelic literature, was born and brought up in Fort William.
 * John McGinlay (born 1964), former Scotland international footballer
 * Mary Mackellar (Màiri NicEalair) (1834–1890), poet and literary translator, was born and brought up in Fort William
 * Justin Ryan, interior decorator and television presenter, although born in Glasgow, was brought up in Fort William.
 * Duncan Shearer (born 1962), former Scotland international footballer
 * Colonel John Wood (1818–1878), recipient of the Victoria Cross for actions during the Anglo-Persian War

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Fort William, Scotland


text:
Talk:Fort William, Scotland

Cromwell
I know the article was short and vague, but - correct me if I'm wrong - I thought Cromwell built a small citadel in the Lochaber region, which was about 50 years later superseded by a stone fort built with the more specific aim of controlling the region in mind. --Dweir

Errr, could be. I know Cromwell built a wooden stockade, and I assumed that was "Fort William" too. I'm not familiar with the circumstances with the changeover to the stone structure. -- Paul Drye

 * Can't find a book on the subject at the moment but the Inverlochy Castle website has Cromwell building his "Citadel" in 1654 and Major-General Hugh Mackay of Scourie raising a fort on the foundations of Cromwell's citadel in 1690.  It was rebuilt in stone over the next four years.
 * There are some excellent plans of Fort William on the National Library of Scotland website . There is also further information in CANMORE on the Royal Commission on the Ancient and Historic Monuments of Scotland site   jmb

Article title
Do we need this level of disambiguation? I wouldn't have thought there were that many Fort Williams in the world. --John (talk) 22:05, 10 October 2012 (UTC)

French claim
"French loan-words in the Gaelic language are very rare" - There are actually quite a few. Not very common, but not uncommon either.--MacRùsgail (talk) 15:16, 6 August 2014 (UTC)

 * Do you think 'rare' by itself is best? Removing 'very' wouldn't find very (ha!) much disagreement from me, but it is a significant fact that French loan words are really not as common in Gaelic as they were in the Norman impact on Anglo-Saxon. Essentially modern English is Norman Anglo-Saxon. Scottish Gaelic is Gaelic with Welsh inflection, and, you are right, more than a few French loan words. Felis Read   (talk) 10:47, 8 August 2014 (UTC)

Lol, you should maybe take a look at the English, Norse and Latin loanwords into Gaelic, as well as Brittonic ones. Gaelic is not some "pure" language anymore than English is. English is just English. Gaelic is just Gaelic. They're not 'essentially' anything other than what they are. — Preceding unsigned comment added by 89.242.79.113 (talk) 20:13, 15 March 2022 (UTC)

Requested move 26 July 2023

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: moved to Fort William, Scotland. (closed by non-admin page mover)  The Night Watch     (talk)   19:37, 2 August 2023 (UTC)

Fort William, Highland → Fort William, Lochaber – Per WP:SCOTLANDPLACE. Convention is to disambiguate settlements in the Highland Council area by their district rather than the council area as a whole. Stevie fae Scotland (talk) 19:31, 26 July 2023 (UTC)
 * Support, there are plenty of Highland & Fort William entries at Highland (disambiguation) and Fort William respectively, so potentially, Highland may be ambiguous dab. Iggy (Swan) (Contribs) 20:43, 26 July 2023 (UTC)
 * I'm also now opting for Fort William, Scotland instead of Lochaber but the reason I gave does not change that much. Iggy (Swan) (Contribs) 17:30, 1 August 2023 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
 * Oppose. Move to Fort William, Scotland. There aren't any other Fort Williams in Scotland. Geographical names should be disambiguated by the most recognizable unambiguous larger administrative unit, which in this case is the country. If the naming convention is promoting absurd titles, ignore it. Policies supersede local guidelines. Recognizability is the primary consideration at Article titles, which is a policy. The naming convention is a local guideline. DrKay (talk) 08:35, 1 August 2023 (UTC)
 * Oppose. Move to Fort William, Scotland. -- Devoke water  12:23, 1 August 2023 (UTC)

Litter on the streets....the bins are full
Empty the bins 2A00:23EE:13D0:6844:FB93:D73D:9B02:53CD (talk) 14:39, 21 April 2024 (UTC)

title: Talk:Fort William, Scotland