Digital Signal Processing Chapter 3 Discrete transforms

2 2.1 Introduction Up to now, we have converted analog signals into discrete-time sequences. So now we can process these sequences by computers. But we cannot process the spectra, i.e., Fourier transforms, in digital way, because they are continuous. In order for that the signals can be processed by computers in the discrete-frequency domain, we should sample the continuous-frequency Fourier transforms, to obtain the discrete-frequency Fourier transforms.

3 3.2 Discrete Fourier transform Sampling X(z) at |z| =1, the discrete Fourier transform can be obtained.

4 3.2 Discrete Fourier transform The continuous-frequency Fourier transform X(e jω ) is sampled by an impulse train as follows This sampling process can be described as According to the convolution theorem, we get that

5 3.2 Discrete Fourier transform

6 This expression shows that, after sampling the continuous-frequency Fourier transform, the sequence becomes periodic. is a precise periodic shifted repetition of x(n).

7 3.2 Discrete Fourier transform For a finite length sequence, its Fourier transform can be represented by the samples in frequency. Only if the number of samples, N, of the Fourier transform in one period is larger than or equal to the length of the sequence, can represent x(n) utterly.

8 3.2 Discrete Fourier transform

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Discrete Fourier transform The expression shows that a finite-length sequence can be obtained from the samples of its Fourier transform. This expression is referred to as the inverse discrete Fourier transform, IDFT for short.

Discrete Fourier transform

Discrete Fourier transform The Fourier transform X(e jω ), after sampled, is defined as the discrete Fourier transform,that is

Discrete Fourier transform

Discrete Fourier transform Usually, the simplified notation is used, that is

Discrete Fourier transform Furthermore, let we get the canonical form of discrete Fourier transforms, that is

Discrete Fourier transform a finite-length sequence with length N: x(n) a periodic sequence with period N: the Fourier transform of x(n): X(e jω ) the Fourier transform of : sampling FT IFT taking one period DFS taking one period a finite-length sequence with length N: x(n) taking one period DFT The discrete Fourier transform of x(n): X(k)

17 Example 1 Compute the DFT of the following sequence Solution

18 Example 1 (cont.) 0 6 0k |H(k)| 0 –π –π/2 π/2 π k arg[H(k)]

19 Example 2 Compute the DFT of the following sequence Solution

20 Example 3 Compute the DFT of the following sequence Solution

21 Comparison between Ex. 2 and Ex

22 Matrix notation of discrete Fourier transform

Properties of the DFT Linearity If x(n) = k 1 x 1 (n) + k 2 x 2 (n), then X(k) = k 1 X 1 (k) + k 2 X 2 (k) Note that the two sequences and two DFTs must have the same length.

Properties of the DFT Time-reversal If x(n) ↔ X(k), then x((– n) mod N) ↔ X ((–k) mod N) x(n)x(n) n x(– n)x(– n) n x((– n) mod N) n

Properties of the DFT Time-shift theorem Note that the sequence x(n+l) should be considered as a circular shift of x(n). x(n)x(n) n x((n–2) mod N) n

Properties of the DFT Circular frequency-shift theorem (modulation theorem) Note that the DFT X(k+l) should also be considered as a circular shift of X(k). From we get that

Properties of the DFT Circular convolution in time

Properties of the DFT Proof

29 Example 1 Compute the 6 point circular convolution. Solution

30 Example 1 (cont.) The 6 point circular convolution is Now, let’s see the linear convolution.

31 Example 1 (cont.) The 6 point circular convolution is and the linear convolution is Comparison between the circular and the linear. –Usually, two convolutions are different from each other; –The circular one is the periodic repetition of the linear one. –In a certain condition, two convolutions could be the same, that is, the period N is greater than or equal to the length of the linear convolution.

Properties of the DFT Circular correlation The DFT of the correlation in time between two sequences is such that Circular autocorrelation

Properties of the DFT Real and imaginary sequences –If x(n) is a real sequence, then Re[X(k)] = Re[X((–k) mod N)] Im[X(k)] = – Im[X((–k) mod N)] –If x(n) is an imaginary sequence, then Re[X(k)] = – Re[X((–k) mod N)] Im[X(k)] = Im[X((–k) mod N)] –If x(n) is a real or an imaginary sequence, then |X(k)| = |X((–k) mod N)| arg[X(k)] = – arg[X((–k) mod N)]

Properties of the DFT Symmetric and antisymmetric sequences –If x(n) = x * ((–n) mod N), it is a periodically conjugate symmetric sequence; –If x(n) = – x * ((–n) mod N), it is a periodically conjugate antisymmetric. Periodically conjugate symmetric and antisymmetric sequences have the following properties: –If x(n) is conjugate symmetric, X(k) is real; –If x(n) is conjugate antisymmetric, X(k) is imaginary;

Properties of the DFT Parseval’s theorem

Properties of the DFT Relation between the DFT and z transform –Sampling the z transform at, we can obtain the DFT. –The z transform can be obtained from the DFT.

Digital filtering using the DFT Linear and circular convolutions Assume that x(n) has duration L and h(n) has duration K. For the linear convolution, the duration of y(n) is L + K –1. If the convolution size N satisfies the condition N ≥ L + K –1 then the circular convolution is equivalent to the linear convolution.

Digital filtering using the DFT Overlap-and-add method … x(n)x(n) LLL h(n)h(n) K L+K–1 y(n)y(n) …

Digital filtering using the DFT Overlap-and-save method … x(n)x(n) h(n)h(n) L+K–1 y(n)y(n) … K L saved discarded K–1

Fast Fourier transform (FFT) Algorithm analysis of DFT When computing the DFT of an N-point sequence, we need about N 2 complex multiplications and N 2 complex additions. In early 1960s, Cooley and Tukey proposed the fast Fourier transform, which requires the order of Nlog 2 N.

Radix-2 decimation-in-time algorithm Separating x(n) into its even- and odd-numbered points, we obtain that or with n=2r for n even and n=2r+1 for n odd,

Radix-2 decimation-in-time algorithm If we note that for N even, then we have that An N-point DFT can be broken into two (N/2)-point computations.

Radix-2 decimation-in-time algorithm Algorithm analysis Computing this DFT, we need complex multiplications and complex additions.

Radix-2 decimation-in-time algorithm If N is equal to a power of 2, we can go on to break two (N/2)-point DFTs into four (N/4)-point DFTs. And repeat this procedure till the computation is reduced to N one point DFTs. For N=2 r, where r is an integer, the procedure of decomposition can be done at most r =log 2 N times.

Radix-2 decimation-in-time algorithm Algorithm analysis once more –When an N-point DFT is broken into two (N/2)-point DFTs, N 2 multiplications are decreased to 2(N/2) 2 +N multiplications; –When an (N/2)-point DFT is decomposed into two (N/4)-point DFTs, (N/2) 2 decreases to 2(N/4) 2 + N/2, so the overall computation needs 2[2(N/4) 2 + N/2]+N = 4(N/4) 2 +2N complex multiplications ; –After carrying out this procedure to the last, we actually accomplish the decompositions r =log 2 N times. At this time, the computation requires Nlog 2 N multiplications.

Radix-2 decimation-in-time algorithm x(0) x(2) x(4) x(6) x(1) x(3) x(5) x(7) X(1) X(2) X(3) X(4) X(5) X(6) X(7) 4-point DFT 4-point DFT X(0)

Radix-2 decimation-in-time algorithm x(0) x(6) x(2) x(4) x(1) x(5) x(3) x(7) X(1) X(2) X(3) X(4) X(5) X(6) X(7) X(0) 2-point DFT 2-point DFT 2-point DFT 2-point DFT

Radix-2 decimation-in-time algorithm x(0) x(6) x(2) x(4) x(1) x(5) x(3) x(7) X(1) X(2) X(3) X(4) X(5) X(6) X(7) X(0)

Radix-2 decimation-in-time algorithm Basic butterfly computation

Radix-2 decimation-in-time algorithm x(0) x(2) x(4) x(6) x(1) x(5) x(3) x(7) X(1) X(2) X(3) X(4) X(5) X(6) X(7) X(0)

Radix-2 decimation-in-time algorithm Several characteristics of decimation-in-time algorithm –In-place –Bit-reversed ordering

Decimation-in-frequency algorithm Dividing X(k) into smaller parts, we can get another class of FFT algorithms.

Decimation-in-frequency algorithm Now consider k even and k odd separately, that is

Decimation-in-frequency algorithm An N-point DFT X(k) can be divided into two (N/2)- point DFTs. Repeat this procedure till an N-point DFT X(k) is divided into N one-point DFTs.

Decimation-in-frequency algorithm X(2) X(4) X(6) X(1) X(3) X(5) X(7) X(0)x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7) 4-point DFT 4-point DFT

Decimation-in-frequency algorithm X(4) X(2) X(6) X(1) X(5) X(3) X(7) X(0)x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7) 2-point DFT 2-point DFT 2-point DFT 2-point DFT

Decimation-in-frequency algorithm X(4) X(2) X(6) X(1) X(5) X(3) X(7) X(0)x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7)

Radix-4 algorithm When N=4 l, we can divide an N-point DFT into four (N/4)-point DFTs.

Radix-4 algorithm

Algorithms for arbitrary values of N If N is not a power of 2 but a composite number, that is N = N 1 N 2 …N l = N 1 N 2l then we can initially divide an N-point DFT into N 1 DFTs with length N 2l. We can continue this decomposition, until the lengths of all DFTs are N l.

Other discrete transforms (skipped)

Signal representations Laplace transform –Time domain: nonperiodic function of a continuous and real time variable. –Frequency domain: nonperiodic function of a continuous and complex frequency variable. z transform –Time domain: nonperiodic function of a discrete and integer time variable. –Frequency domain: nonperiodic function of a continuous and complex frequency variable.

Signal representations Fourier transform of continuous signals –Time domain: nonperiodic function of a continuous and real time variable. –Frequency domain: nonperiodic function of a continuous and imaginary frequency variable. Fourier transform of discrete signals –Time domain: nonperiodic function of a discrete and integer time variable. –Frequency domain: periodic function of a continuous and imaginary frequency variable.

Signal representations Fourier series of periodic continuous signals –Time domain: periodic function of a continuous and real time variable. –Frequency domain: nonperiodic function of a discrete and integer frequency variable. Fourier series of periodic discrete signals (DFS) –Time domain: periodic function of a discrete and integer time variable. –Frequency domain: periodic function of a discrete and integer frequency variable.

Signal representations Discrete Fourier transform (DFT) –Time domain: finite-length function of a discrete and integer time variable. –Frequency domain: finite-length function of a discrete and integer frequency variable. Fast Fourier transform (FFT) –FFT is no transform –FFT is a kind of fast algorithm of DFT

66 Exercises 3.3 (d) (b)(c) 3.17