1 Chapter 8 The Discrete Fourier Transform (cont.)

2 Introduction  Last week, we have discussed the following issues:  Periodic sequences can be represented using discrete Fourier series (DFS);  The coefficients of DFS are the same as the Fourier transform at equally spaced frequency samples;  Therefore, for periodic sequences, sampling of frequency response becomes possible;  When the DFS is used to represent finite-length sequence, it becomes discrete Fourier transform (DFT).  Today, we will discuss the properties of DFT, and the linear convolution using the DFT.

3 Properties of DFT (1)  Note the analogy and difference between the properties of DFT and properties of DFS. Linearity  If two finite-duration sequences x 1 [n] and x 2 [n] are linearly combined, i.e., if then its DFT is  Clearly, when x 1 [n] has length N 1, and x 2 [n] has length N 2, the maximum length of x 3 [n] is N 3 =max[N 1, N 2 ]. For the above DFT to be meaningful, both DFTs must be computed with the same length N ≥ N 3. [In DFS, two sequences should have the same periods.]

4 Properties of DFT (2) Circular Shift of a Sequence  If, we want to find x 1 [n] such that Since both x[n] and x 1 [n] must be zero outside the interval 0 ≤ n ≤ N – 1, x 1 [n] cannot result from a simple time shift of x[n].  The correct result follows directly from the result of DFS and the interpretation of DFT.

5 Properties of DFT (3) Circular Shift of a Sequence - Example

6 Properties of DFT (4) Duality  Consider x[n] and its DFT X[k]. Let From the duality property of DFS  If we define the periodic sequence, one period of which is the finite-length x 1 [n] = X[n], then the DFS coefficients of are. Therefore, the DFT of x 1 [n] = X[n] is  The sequence Nx[((–k )) N ] is Nx[k] index reversed, modulo N.

7 Properties of DFT (5) Duality - Example

8 Properties of DFT (6) Symmetric Properties  Define periodic conjugate-symmetric (periodic even if real) periodic conjugate-antisymmetric (periodic odd if real)  It can be shown that x ep [n] and x op [n] can be generated by aliasing x e [n] and x o [n] into the interval 0 ≤ n ≤ N – 1.

9 Compare: DFS  If, then Properties of DFT (7) Symmetric Properties (cont.)  If, then

10 Properties of DFT (8) Symmetric Properties (cont.)  When x[n] is real, then Compare: DFS  When is real, then

11 Properties of DFT (9) Circular Convolution (compare: Periodic Convolution in DFS)  Consider two finite-duration sequences x 1 [n] and x 2 [n], both of length N, with DFTs X 1 [k] and X 2 [k], respectively. We wish to determine the sequence x 3 [n] whose DFT is X 3 [k]=X 1 [k] X 2 [k].  Use the result of DFS, we have

12 Properties of DFT (10) Circular Convolution (cont.)  For the convolution defined here, the second sequence is circularly time reversed and circularly shifted with respect to the first. For this reason, we call this operation circular convolution, or more specifically, N-point circular convolution, explicitly identifying the fact that both sequences have length N (or less) and that the sequences are shifted modulo N.  The operation of forming a sequence x 3 [n] as the N-point circular convolution of x 1 [n] and x 2 [n] are denoted as x 3 [n] = x 1 [n] x 2 [n] = x 2 [n] x 1 [n]  In view of the duality of the DFT relations, it is not surprising to see that x 1 [n]x 2 [n]

13 Properties of DFT (11) Circular Convolution - Example [Circular Convolution with a Delayed Impulse]

14 Properties of DFT (12) Circular Convolution - Example [Circular Convolution of Two Rectangular Impulses] (N=L=6)

15 Properties of DFT (13) Circular Convolution - Example [Circular Convolution of Two Rectangular Impulses] (N=2L=12)

16 Linear Convolution using DFT (1)  Because efficient algorithms of calculating DFT are available (Chapter 9), it is computationally efficient to implement a convolution of two sequences by the following procedure.  Compute the N-point DFTs X 1 [k] and X 2 [k] of the two sequences x 1 [n] and x 2 [n], respectively.  Compute the product X 3 [k]=X 1 [k] X 2 [k] for 0 ≤ n ≤ N – 1.  Compute the sequence x 3 [n] = x 1 [n] x 2 [n] as the inverse DFT of X 3 [k].  Consider two sequences x 1 [n] (of length L) and x 2 [n] (of length P), then the product x 1 [m]x 2 [n–m] is zero for all m whenever n L+P–2. That is, x 3 [n] ≠0 for 0 ≤ n ≤ L+P–2. Therefore, (L+P–1) is the maximum length of the sequence obtained from their linear convolution.

17 Linear Convolution using DFT (2)

18 Circular Convolution as Linear Convolution with Aliasing (1)  We have seen from the previous two examples that, whether a circular convolution is the same as the linear convolution of the corresponding finite-length sequence depends on the length of the DFT in relation to the length of the finite-length sequence.  A very useful interpretation of the relationship between them is in terms of time aliasing.  We have observed that if the Fourier transform X(e j  ) of a sequence x[n] is sampled at frequencies  k =2  k/N, then the resulting sequence corresponds to the DFS coefficients of the periodic sequence (r is an integer)

19 Circular Convolution as Linear Convolution with Aliasing (2)  From our discussion of the DFT, we know that is the DFT of one period of ; i.e., the subscript p is used to denote that a sequence is one period of a periodic sequence resulting from an inverse DFT of a sampled Fourier transform.  If x[n] has length less than or equal to N, no time aliasing occurs and x p [n] = x[n]. However, if the length of x[n] is greater than N, x p [n] may not equal to x[n] for some or all values of n.

20 Circular Convolution as Linear Convolution with Aliasing (3)  Consider two sequences x 1 [n] and x 2 [n], with Fourier transforms are X 1 (e j  ) and X 2 (e j  ), respectively. The sequence x 3 [n] = x 1 [n]*x 2 [n] has the Fourier transform X 3 (e j  ) = X 1 (e j  ) X 2 (e j  ).  If we define a DFT Therefore,  The sequence resulting as the inverse DFT of X 3 [k] is

21 Circular Convolution as Linear Convolution with Aliasing (4)  Therefore, the circular convolution of two finite-length sequences is equivalent to linear convolution of the two sequences, followed by time aliasing.  Note that if N is greater than or equal to either L or P, X 1 [k] and X 2 [k] represent x 1 [n] and x 2 [n] exactly. But x 3p [n] = x 3 [n] for all n only if N is greater than or equal to the length of the sequence x 3 [n], which is (L+P–1).  Therefore, the circular convolution corresponding to X 1 [k] X 2 [k] is identical to the linear convolution x 1 [n] * x 2 [n] if N, the length of the DFTs, satisfies N ≥ (L+P–1).

22 Circular Convolution - Example [Circular Convolution of Two Rectangular Impulses] Circular Convolution as Linear Convolution with Aliasing (5)

23 Circular Convolution as Linear Convolution with Aliasing (6)  When N=L=P, all of the sequence values of the circular convolution may be different from those of the linear convolution.  If P < L, some of the sequence values in an L-point circular convolution will be equal to the corresponding sequence values of the linear convolution.  Next, we show some graphical illustration.

24 Example of linear convolution of two finite-length sequences Circular Convolution as Linear Convolution with Aliasing (7)

25 Illustration of aliasing for different N Circular Convolution as Linear Convolution with Aliasing (8)

26 Implementing LTI Systems Using the DFT (1)  We consider LTI systems which can be implemented by convolution. This implies that the circular convolution can be used to implement these systems.  First, we consider an L-point input sequence x[n] and a P-point impulse response h[n]. The linear convolution of these two sequence, y[n], has finite duration with length (L+P–1).  The DFTs that we compute must be of at least that length. Therefore, both x[n] and h[n] must be augmented with sequence values of zero magnitude. This process is often referred to as zero-padding.

27 Implementing LTI Systems Using the DFT (2)  In many applications, the input signal is of indefinite duration (speech waveform, communication signals…).  Theoretically, we might be able to store the entire waveform and then implement the DFT procedure, such a DFT is generally impractical to compute. Another consideration is that for this method of filtering, no filtered samples can be computed until all the input samples have been collected.  To avoid the two problems, the solution is to use block convolution, in which the signal can be filtered is segmented into sections of length L. Each section can then be convolved with the finite-length impulse response and the filtered sections fitted together in an appropriate way. The linear filtering of each block can then be implemented using the DFT.

28 Implementing LTI Systems Using the DFT (3)  We assume that x[n]=0 for n<0 and that the length of x[n] is much greater than P. We can represent x[n] as a sum of shifted- length segments of length L; i.e., with

29 Implementing LTI Systems Using the DFT (4)  Because convolution is a LTI operation, we have with  Note that each term of y r [n] has length (L+P–1). Thus, the linear convolution x r [n] * h[n] can be obtained by using N-point DFTs, where N ≥ (L+P–1).  Since the beginning of each input section is separated from its neighbors by L points whereas each filtered section has length (L+P–1), the nonzero points in the filtered sections will overlap by (P–1) points. These overlap samples must be added in carrying put the sum required by the above equations.  This procedure is often referred as the overlap-add method.

30 Implementing LTI Systems Using the DFT (5)

31 Implementing LTI Systems Using the DFT (6)  An alternative block convolution procedure, commonly called the overlap-save method, corresponds to  implementing an L-point circular convolution of a P-point impulse response h[n] with an L-point segment x r [n] and  identifying the part of the circular convolution that corresponds to a linear convolution.  When P < L, the first (P–1) points of the circular convolution result are incorrect, while the remaining points are identical to those of linear convolution.  We divide x[n] into sections of length L so that each input section overlaps the preceding section by (P–1) points. That is,

32 Implementing LTI Systems Using the DFT (7)

33 Implementing LTI Systems Using the DFT (8)  The circular convolution of each section with h[n] is denoted y rp [n], the extra subscript p indicating that y rp [n] is the result of a circular convolution in which time aliasing has occurred. The portion of each output section in the region 0 ≤ n ≤ P – 2 must be discarded.  The remaining samples from successive sections are then abutted to construct the final filtered output. That is with

34 Summary  Properties of DFT  Linearity  Circular shift of a sequence  Duality  Symmetric properties  Circular Convolution  Linear Convolution using DFT  Circular convolution vs. linear convolution  Implementing LTI Systems Using the DFT  Overlap-add method  Overlap-save method

35 Homework Assignments (11)