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1 DIGITAL SIGNAL PROCESSING DEPT./SEM.: ECE&EEE /V DISCRETE FOURIER TRANFORM AND FFT PART-A 1. Define DFT of a discrete time sequence? AUC MAY 06 The DFT is used to convert a finite discrete time sequence x(n) to an N-point frequency domain sequence denoted by X(k),. The N-point DFT of a finite duration sequence x (n) of length L, where L N is defined as N-1 X(K) = x(n)e ^(-j2 nk/n) ; for k = 0,1,2,.,(N - 1) N=0 2. Define IDFT? AUC APR 05 The IDFT is used to convert the N-point frequency domain sequence X(k) to an N-point time domain sequence. The IDFT of the sequence X(k) of length N is defined as N-1 X(n) = x(k)e ^(-j2 nk/n) ; for n = 0,1,2,.,(N - 1) N=0 3. What is the relation between DTFT and DFT? AUC NOV 11 Let x(n) be a discrete time sequence. Now DTFT {x(n)} = X( ) or FT{x(n)} X( ) and DFT {x(n)} X(k). The X( ) is a periodic continuous function of and X(k) is an N- point periodic sequence. The N-point sequence X(k) is actually N samples of X( ) which can be obtained by sampling one period of X( ) at N equal intervals. 4. What is the drawback in Fourier Transform and how it is overcome? AUC NOV 12 The drawback in Fourier transform is that it is a continuous function of and so it cannot be processed by digital system. This drawback is overcome by using Discrete Fourier Transform. The DFT converts the continuous function of.

2 5. Give any two applications of DFT (or Mention the importance of DFT) AUC NOV The DFT is used for spectral analysis of signals using a digital computer. 2. The DFT is used to perform filtering operations on signals using digital computer. 6. when an N-Point periodic sequence is said to be even or odd sequence? AUCMAY 11 An N-Point periodic sequence is called even if it satisfies the condition x(n-n) = x(n); for0 n (N-1) An N-Point periodic sequence is called odd if it satisfies the condition x(n-n) = - x(n); for0 n (N-1) 7. List any four properties of DFT. AUC NOV 09 Let DFT {x(n)} =X(k),DFT{x 1 (n)} = X 1 (k)and DFT{x 2 (k)} = X 2 (k) i. Periodicity: X(K+N) = X(K); for all k ii Linearity : DFT { a 1 x 1 (n)+a 2 x 2 (n)} = a 1 x 1 (k)+a 2 x 2 (k), where a 1 & a 2 are constant iii DFT of time reversed sequence : DFT {x(n-n)} = X (N-k) iv Circular convolution : DFT {x 1 (n) * X 2 (n) = X 1 (k)x 2 (k) 8. Why linear convolution is important in DSP? AUC NOV 05 The response or output of LTI discrete time system for any input x(n) is given by linear convolution of the input x(n) and the impulse response h(n) of the system. (This means that if the impulse response of a system is known, then the response of the system for any input can be determined by convolution operation). 9. Write the properties of linear convolution? AUC APR 07, 08 The linear convolution satisfies the following properties. I. Commutative property : x(n)*h(n) = h(n)*x(n) II. Associative property : [x(n)*h 1 (n)]* h 2 (n) = x(n)*[h 1 (n)*h 2 (n)] III. Distributive property : [x(n)*[h 1 (n)]+h 2 (n) ]= x(n)*h 1 (n)+x(n)*h 2 (n)

3 10. List the differences between linear convolution and circular convolution? AUC NOV 09 Linear convolution 1. The length of the input sequence can be different. 2. Zero padding is not required Circular convolution 1. The length of the input sequence should be same. 2. If the length of the input sequence are different, then zero padding is required. 3. The input sequences need not be period. 4. The output sequence is nonperiodic. 5. The length of output sequence will be greater than the length of input sequences. 3. Atleast one of the input sequence should be periodic or should be periodically extended. 4.The output sequence is periodic. The periodicity is same a that of input sequence. 5. The length of the input and output sequences are same. 11. What is radix-2 FFT? AUC MAY 05 The radix 2 FFT is an efficient algorithm for computing N point DFT of an N-point sequence. In radix 2 FFT the N-point sequence is decimated into 2-point sequences and the 2-point DFT for each decimated sequence is computed. From the results of 2-point DFTs, the 4-point DFTs are computed. From the results of 4-point DFTs, the 8-point DFTs are computed and so on until we get N-point DFT. 12. How many multiplications and additions are involved in radix-2 FFT? AUC NOV 06 For performing radix 2 FFT, the value of N should be such that, N = 2 m. The total number of complex additions are Nlog 2 N and the total number of complex multiplications are (N/2) log 2 N. 13. What is DIT radix 2 FFT? AUC APR 08 The DIT (Decimation In Time) radix-2 FFT is an efficient algorithm for computing DFT. In DIT radix-2 FFT, the time domain N-point sequence is decimated into 2-point sequences.

4 The result of 2-point DFTs are used to compute 4-point DFTs. Two numbers of 2-point DFTs are combined to get a 4-point DFT. The results of 4-point DFTs are used to compute 8 point DFTs. Two numbers of 4-point DFTs are combined to get an 8-point DFT. This process is continued until we get N-point DFT. PART-B 1.Explain the properties of DFT. AUC NOV 06, Periodicity 2. Linearity 3. Symmetry property. 4. Circular convolution of two sequences. 5. Time reversal of sequences. 6. Circular time shift of a sequence. 7. Circular frequency shift of a sequence. 8. Circular correlation of two sequences. 9. Multiplication of two sequences. 10. Parsevals theorem 1. Periodicity: X(n+N)=X(n) X(K+N)=X(K) 2. Linearity: Dft ax 1(n)+ bx 2 (n) N point ax 1(n)+ bx 2 (n) 3. Symmetry property: x(n) Dft N point X(k) x * (n) DFT X(N-K) N-point

5 4. Circular convolution of two sequences. x 1 (n) x 2 (n) X 1 (K). X 2 (K) 5. Time reversal of a sequence: Dft x(n) Npoint DFT x(-n) N= x(n-n) X(-K) N = X(N-K) N-point X(k) 6. Circular time shift of a sequence: x(n) Dft Npoint X(k) x(n-m) Dft Npoint -j2 π mk / N X(k) e 7. Circular frequency shift of a sequence: Dft x(n) Npoint X(k) j2 π mn / N x(n) e X(k-m) Dft Npoint N

6 8. Circular correlation of two sequences. N-1 x 1 (n) x 1 * (n-m) X 1 (k) X 2 * (k) n=0 9. Multiplication of two sequences. x 1 (n) x 2 (n) DFT 1/N [X 1 (k). X 2 * (k)] 10. Parsevals theorem N-1 N-1 x 1 (n) 2 1/N X(k) 2 n=0 n=0 2.Derive and draw the FFT algorithm radix 2 DIT algorithm. AUC MAY 07, 09 Decimation in time algorithm In this case, let us assume that x(n) represents a sequence of N values, where, N is an integer power of 2, that is N=2 L. The given sequence is decimated (broken) in to two n/2 point sequences consisting of the even numbered values of x(n) and the odd numbered values of x(n). The N- point DFT sequence x(n) is given by, N-1 nk X(K) = x(n) W N 0 k N-1 n=0 breaking x(n) in to its even and odd numbered values, we obtain,

7 X(K) = x(n) W N nk + x(n) WN nk n=0 n=0 n=even n=odd sub n=2r for n even and n=2r+1 for n is odd, we have, X(K) = 2rk x(2r) W N + (2r+1)k x(2r+1) WN r=0 r=0 X(K) = x(2r) W N 2rk + x(2r+1) WN (2rk) W N K r=0 r=0 X(K) = rk x(2r) W N/2 + rk x(2r+1) WN/2 K WN r=0 r=0 X(K)=G(K) + W N K H(K) where K=0,1,2,3,4 Where G(K) and H(K) are the N/2 points DFTS of the even and odd numbered sequence respectively. Here each of the sum is computed for 0 k

8 For the direct computation of an N- point dft, G(K) and H(K) as a combination of two n/2 points DFT rk G(K) = x(2r) W N/2 r=0 sub r = 2l for n even and r = 2l+1 for n is odd, we have, N/4-1 N/4-1 G(K) = 2lk x(2l) W N/2 + (2l+1)k x(2l+1) WN/2 l=0 l=0 N/4-1 N/4-1 G(K) = x(2l) W N/4 lk + x(2l+1) WN/4 lk WN/2 K l=0 l=0 G(K)=G 1 (K)+ W N/2 K G 2 (K)

9 Similarly; H(K) = rk x(2r+1) W N/2 r=0 sub r = 2l for n even and r = 2l+1 for n is odd, we have, N/4-1 H(K) = x(4l+1) W N/2 2lk N/4-1 + x(2(2l+1)+1) WN/2 (2l+1)k l=0 l=0 N/4-1 N/4-1 G(K) = x(4l+1) W N/4 lk + x(4l+3) WN/4 lk WN/2 K l=0 l=0 H(K)=H 1 (K)+ W N/2 K H 2 (K) A+BW N 1 A-BW N 1

10 1 3. Derive and draw the FFT algorithm radix 2 DIF algorithm. AUC NOV 07, 08 Decimation in frequency algorithm (DIF-FFT): In the DIT-FFT algorithm, the time domain sequence x(n) is divided in to smaller subsequence.in this algorithm,the frequency samples X(K) are divided in to smaller and smaller subsequences. W.K.T N-1 nk X(K) = x(n) W N 0 k N-1 n=0 X(K) = nk x(n) W N + nk x(n) WN n=0 n=0

11 sub n=n+n/2 X(K) = nk x(n) W N + n+n/2)k x(n+n/2) WN n=0 n=0 X(K) = x(n) W N nk + x(n+n/2) WN NK/2 WN Kn here, n=0 n=0 W N NK/2 =(-1) k X(K) = x(n) W N nk + x(n+n/2) (-1) k W N Kn equ I n=0 n=0 in the above equation Put k=2r, when k is even X(2r) = x(n) W N 2r n + x(n+n/2) (-1) 2r W N 2r n n=0 n=0 X(2r) = n=0 [ x(n) + x(n+n/2) ] W N 2r n X(2r) = [ x(n) +x(n+n/2) ] W N/2 r n n=0 put k=2r +1, when k is odd,

12 (2r+1) n X(2r+1) = x(n) W N + x(n+n/2) (-1) (2r+1) W N (2r+1) n n=0 n=0 X(2r+1) = [ x(n) - x(n+n/2) ] W N (2r+1) n n=0 X(2r+1) = [ x(n) - x(n+n/2) ] W N/2 r n n=0 WN n r n X(2r) = g(n) W N/2 n=0 X(2r+1) = h(n) W N/2 r n n=0 WN n g(n) = [ x(n) - x(n+n/2) ] h(n) =[ x(n) + x(n+n/2) ]

13 A+B (A-B)W N 0

14 4.find the DFT of a sequence X(N)={1,2,3,4,4,3,2,1} by using DIT- FFT algorithm. Solution: W 8 0 =1 W 8 1 =1/ 2 - j 1/ 2 W 8 2 = -j W 8 3 = - 1/ 2 - j 1/ 2

15 5.find the DFT of a sequence X(N)={1,2,3,4,4,3,2,1} by using DIF- FFT algorithm. AUC MAY 08, 06 Solution: W 8 0 =1 W 8 1 =1/ 2 - j 1/ 2 W 8 2 = -j W 8 3 = - 1/ 2 - j 1/ 2

16 6. Discuss about linear filtering in FFT? AUC NOV 06,MAY 09 Linear filtering by fft : Overlap save method: For the input sequence x(n) which has a long duration, performing convolution in practical is not possible. therefore, the entire sequence is divided in to blocks. each blocks are separately and finally the result are combined. The o/p response obtained by this method will be same as that of linear convolution. two methods are used in filtering, they are overlap- save and add method. Overlap save method: Let the length of the input sequence is L S length of the impulse response is M. now the sequence is divided in to blocks of sign N=L-M-1 Each block will have last (M-1) data points of previous block followed by new data points. For the first of data the first (M-1) points are set to zero. X 1 (n)= 0,0, x(0),x(n) x(l-1)

17 (M-1) zero X 2 (N)= X(L-M+1).x(L-1), x(l).x(2l-1) (m-1) data points from x(n) L new data point NOW the impulse response of FIR filter is increased by length appending (L-1) zeroes and N-point circular convolution of x 1 (n) and h(n) are computed. Y 1 (n)=x 1 (n) O h(n) Discard the first M-1 points of the filtered sections x 1 (n) O h(n) Overlap add method: Let the length of the sequence is L s length of the impulse response is M. now the sequence is divided in to blocks of data single having length L and (M-1) zeros are appending to it to make the data single of N=L+M-1 Thus the data blocks are represented as X 1 (n)= x(0),x(1) x(l-1) 0,0, (M-1) zero appended X 2 (n)= X(L),X(L+1).x(2L-1), 0,0,0,. Now L-1 zeros are added to the impulse response (h(n)) N-point circular convolution is performed. since each data block is terminated with (M-1) zeros, last (M-1) point from each output block must be overlapped and added to the first (L-1) points of the succeeding block. Hence this method is called overlap add method. Let the o/p blocks are of the form, Y 1 (n)= { y 1 (0),y 1 (1),..y 1 (L-1), y 1 (1) y 1 (L-1)} Y 2 (n)= { y 2 (0),y 2 (1),..y 2 (L-1), y 2 (1) y 2 (L-1)}

18 Y 3 (n)= { y 3 (0),y 3 (1),..y 3 (L-1), y 3 (1) y 3 (L-1)} The output sequence is y(n)={ y 1 (0), y 1 (1),. y 1 (L-1),y 1 (L)+ y 2 (0),. Y 1 (N-1)+ y 2 (M-1),y 2 (M). y 2 (L)+ y 3 (0), y 2 (L+1)+ y 3 (1),.. y 3 (N-1)} 7. Calculate the IDFT of the sequence X(K)={4,0,0,0}. AUC NOV 05 SOLUTION: X(K) X(K)={4,0,0,0} x(0) x(1) 1 -j -1 j 0 = 1/N x(2) x(3) 1 j -1 -j 0 4 = 1/

19 1 = X(n)=( 1,1,1,1) 8. Calculate the DFT of the sequence x(n)={1,2,3,4}. AUC MAY 08 X(0) X(1) 1 -j -1 j 2 = X(2) X(3) 1 j -1 -j = 1-2j-3+4j j-3-4j

20 10 = -2+2j j