程序员专用 DSP 算法：数学原理与源码实现

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程序员专用 DSP 算法：数学原理与源码实现（(Dsp) Algorithms For Programmers）

摘要：《(Dsp) Algorithms For Programmers》是一本面向程序员的 DSP 算法与数学原理实战手册，聚焦数字信号处理核心算法的数学原理与代码实现。全书系统讲解离散傅里叶变换、基 2 / 基 4 / 分裂基 FFT、卷积、Z 变换、哈特利变换、数论变换、沃尔什变换、哈尔变换等核心算法，给出递归与非递归实现、三角函数计算优化、内存访问改进及代码自动生成方法。同时涵盖位操作、排列、排序、组合算法、高精度运算等底层技术，提供可直接复用的伪代码与 C++ 源码，兼顾理论推导与工程效率，适合 DSP 开发、算法实现与高性能计算参考。

Algorithms for programmers ideas and source code

This document is work in progress: read the ”important remarks” near the beginning

J¨org Arndt

arndt@jjj.de

This document1 was LATEX’d at September 26, 2002

Some important remarks

about this document.

This draft is intended to turn into a book about selected algorithms. The audience in mind are pro- grammers who are interested in the treated algorithms and actually want to have/create working and reasonably optimized code.

The printable full version will always stay online for free download. It is planned to also make parts of the TEXsources (plus the scripts used for automation) available. Right now a few ﬁles of the TEX sources and all extracted pseudo-code snippets1 are online. The C++-sources are online as part of FXT or hfloat (arithmetical algorithms).

The quality and speed of development does depend on the feedback that I receive from you. Your criticis m concerning language, style, correctness, omissions, technicalities and even the goals set here is very welcome. Thanks to those2 who helped to improve this document so far! Thanks also to the people who share their ideas (or source code) on the net. I try to give due references to original sources/authors wherever I can. However, I am in no way an expert for history of algorithms and I pretty sure will never be one. So if you feel that a reference is missing somewhere, let me know.

New chapters/sections appear as soon as they contain anything useful, sometimes just listings or remarks outlining what is to appear there.

A ”TBD: something to be done” is a reminder to myself to ﬁll in something that is missing or would be nice to have.

The style varies from chapter to chapter which I do not consider bad per se: while some topics (e.g. FFTs) need a clear and explicit introduction others (e.g. the bitwizardry chapter) seem to be best presented by basically showing the code with just a few comments. Still other parts (e.g. sorting) are presented elsewhere extremely well so I will introduce the basic ideas only very shortly and supply some (hopefully) useful code.

Sprache will partly go away: using/including the actual code from FXT will be beneﬁcial to both this document and FXT itself. The goal is to automatically include the functions referenced. Clearly, this will drastically reduce the chance of errors in the shown code (and at the same time drastically reduce the workload for me). Initially I planned to write an interpreter for Sprache, it just never happened. At the same time FXT will be better documented which it really needs. As a consequence Sprache will only be used when there is a clear advantage to do so, mainly when the corresponding C++ does not appear to be self explanatory. Larger pieces of code will be presented in C++. A tiny starter about C++ (some good reasons in favor of C++ and some of the very basics of classes/overloading/templates) will be included. C programmers do not need to be shocked by the ζ++’: only an rather minimal set of the C++ features is used.

The theorem-like environment for the codes shall completely go away. It leads to duplication of state- ments, especially with non-pseudo code (running text, description in the environment and comments at the begin of the actual code).

Enjoy reading !

List of important Symbols

沦x real part of x

$x imaginary part of x

x∗ complex conjugate of x

a a sequence, e.g. {a0 , a1 , ..., an−1}, the index always starts with zero.

a transformed (e.g. Fourier transformed) sequence

m emphasize that the sequences to the left and right are all of length m =

F [a] (= c) (discrete) Fourier transform (FT) of a, ck = Σx(n) 0(1) ax zx k where z = e±2π i/n

F−1 [a] inverse (discrete) Fourier transform (IFT) of a, F−1 [a]k = Σx(n) 0(1) ax z −x k

Ska a sequence c with elements cx := ax e± k 2π ix/n

H [a] discrete Hartley transform (HT) of a

a sequence reversed around element with index n/2

aS the symmetric part of a sequence: aS := a + a

aA the antisymmetric part of a sequence: aA := a − a

Z [a] discrete z-transform (ZT) of a

Wv [a] discrete weighted transform of a, weight (sequence) v

W 1 [a] inverse discrete weighted transform of a, weight v

a ④ b cyclic (or circular) convolution of sequence a with sequence b

a ④ac b acyclic (or linear) convolution of sequence a with sequence b

a ④ − b negacyclic (or skew circular) convolution of sequence a with sequence b

a ④{v} b weighted convolution of sequence a with sequence b, weight v

a ④⊕ b dyadic convolution of sequence a with sequence b

n\N n divides N

n ⊥ m g cd(n, m) = 1

a(j%m) sequence consisting of the elements of a with indices k: k ≡ j mod m e.g.

a(even) , a(odd)a(0%2) , a(1%2)

a(j/m) sequence consisting of the elements of a with indices k: j · n/m ≤ k < (j + 1) · n/m e.g.

a(left) , a(right)a(0/2) , a(1/2)

Chapter 1 The Fourier transform

1.1 The discrete Fourier transform

The discrete Fourier transform (DFT or simply FT) of a complex sequence a of length n is deﬁned as

z is an n-th root of unity: zn = 1.

Backtransform (or inverse discrete Fourier transform IDFT or simply IFT) is then

To see this, consider element y of the IFT of the FT of a:

As Σk (z①__y )k = n for x = y and zero else (because z is an n-th root of unity). Therefore the whole

expression is equal to

where

Here we will call the FT with the plus in the exponent the forward transform. The choice is actually arbitrary1 .

CHAPTER 1. THE FOURIER TRANSFORM

The normalization factor in front of the FT sums is sometimes replaced by a single in front of the inverse FT sum which is often convenient in computation. Then, of course, Parseval’s equation has to be modiﬁed accordingly.

A straight forward implementation of the discrete Fourier transform, i.e. the computation of n sums each of length n requires ~ n2 operations:

[FXT: slow ft in slow/slowft.cc] is must be +1 (forward transform) or __1 (backward transform), SinCos(x) returns a Complex(cos(x), sin(x)).

A fast Fourier transform (FFT) algorithm is an algorithm that improves the operation count to propor- tional n Σ k __ 1), where n = p1p2 · · · pm is a factorization of n. In case of a power n = pm the value computes to n (p __ 1) logp (n). In the special case p = 2 even n/2 log2 (n) (complex) multiplications suﬃce. There are several diﬀerent FFT algorithms with many variants.

1.2 Symmetries of the Fourier transform

A bit of notation turns out to be useful:

Let a be the sequence a (length n) reversed around element with index n/2:

Let aS , aA be the symmetric, antisymmetric part of the sequence a, respectively

(The elements with indices 0 and n/2 of aA are zero). Now let a ∈ R (meaning that each element of a is ∈ R), then

The FT of a real symmetric sequence is real and symmetric and the FT of a real antisymmetric sequence is purely imaginary and antisymmetric. Thereby the FT of a general real sequence is the complex conjugate of its reversed:

Similarly, for a purely imaginary sequence b ∈ iR:

The FT of a complex symmetric/antisymmetric sequence is symmetric/antisymmetric, respectively.

1.3 Radix 2 FFT algorithms

1.3.1 A little bit of notation

Always assume a is a length-n sequence (n a power of two) in what follows:

Let a(even) , a(odd) denote the (length-n/2) subsequences of those elements of a that have even or odd indices, respectively.

Let a(left) denote the subsequence of those elements of a that have indices 0 . . . n/2 __ 1.

Similarly, a(right) for indices n/2 n __ 1.

Let S ka denote the sequence with elements ax e§ k 2π ix/n where n is the length of the sequence a and the sign is that of the transform. The symbol S shall suggest a shift operator. In the next two sections only S 1/2 will appear. S0 is the identity operator.

1.3.2 Decimation in time (DIT) FFT

The following observation is the key to the decimation in time (DIT) FFT2 algorithm: For n even the k-th element of the Fourier transform is

where z = e§i 2π/n and k ∈ {0, 1, , n __ 1}.

The last identity tells us how to compute the k-th element of the length-n Fourier transform from the length-n/2 Fourier transforms of the even and odd indexed subsequences.

To actually rewrite the length-n FT in terms of length-n/2 FTs one has to distinguish the cases 0 ≤ k < n/2 and n/2 ≤ k < n, therefore we rewrite k ∈ {0, 1, 2, . . . , n __ 1} as k = j + δ where j ∈ {0, 1, . . . , n/2 __ 1}, δ ∈ {0, 1}.

Notingthatz2 is just therootofunitythatappears ina length-n/2FTonecanrewritethe last two equationsasthe

Idea1.1(FFTradix2DITstep)Radix2decimationintimestepfortheFFT:

(Here it is silently assumed that ’+’ or ’__’ between two sequences denotes elementwise addition or subtraction.)

The length-n transform has been replaced by two transforms of length n/2. If n is a power of 2 this scheme can be applied recursively until length-one transforms (identity operation) are reached. Thereby the operation count is improved to proportional n · log2 (n): There are log2 (n) splitting steps, the work in each step is proportional to n.

Code 1.1 (recursive radix 2 DIT FFT) Pseudo code for a recursive procedure of the (radix 2) DIT FFT algorithm, is must be +1 (forward transform) or -1 (backward transform):

The data length n must be a power of 2. The result is in x []. Note that normalization (i.e. multiplication of each element of x [] by 1/√n) is not included here.

[FXT: recursive dit2 fft in slow/recfft2.cc] The procedure uses the subroutine

Code 1.2 (Fourier shift) For each element in c[0..n-1] replace c[k] by c[k] times ev 2 π ik/n . Used with U = § 1/2 for the Fourier transform.

The recursive FFT-procedure involves n log2(n) function calls, which can be avoided by rewriting it in

a non-recursive way. One can even do all operations in place, no temporary workspace is needed at

all. The price is the necessity of an additional data reordering: The procedure revbin_permute(a[],n)

rearranges the array a[] in a way that each element ax is swapped with a˜x, where ˜x is obtained from x

by reversing its binary digits. This is discussed in section 8.1.

Code 1.3 (radix 2 DIT FFT, localized) Pseudo code for a non-recursive procedure of the (radix 2)

DIT algorithm, is must be-1 or +1:

[FXT: dit2-fft-localized in fft/fftdit2.cc]

This version of a non-recursive FFT procedure already avoids the calling overhead and it works in place. It works as given, but is a bit wasteful. The (expensive!) computation e := exp(is*2*PI*I*j/m) is done n/2 · log2 (n) times. To reduce the number of trigonometric computations, one can simply swap the two inner loops, leading to the ﬁrst ζreal world’ FFT procedure presented here:

Code 1.4 (radix 2 DIT FFT) Pseudo code for a non-recursive procedure of the (radix 2) DIT algo- rithm, is must be -1 or +1:

procedure fft_dit2(a [] , ldn, is)

// complex a[0 . .2**ldn-1] input, result

Swapping the two inner loops reduces the number of trigonometric (exp()) computations to n but leads to a feature that many FFT implementations share: Memory access is highly nonlocal. For each recursion stage (value of ldm) the array is traversed mh times with n/m accesses in strides of mh. As mh is a power of 2 this can (on computers that use memory cache) have a very negative performance impact for large values of n. On a computer where the CPU clock (366MHz, AMD K6/2) is 5.5 times faster than the memory clock (66MHz, EDO-RAM) I found that indeed for s mall n the localized FFT is slower by a factor of about 0.66, but for large n the same ratio is in favour of the ζnaive’ procedure!

It is a good idea to extract the ldm==1 stage of the outermost loop, this avoids complex multiplications with the trivial factors 1 + 0 i: Replace

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