Digital Time Signal Processing

BE EXTC D T S P DEC- 2004 By Kiran Talele ( talelesir@yahoo.com ) Q 1. (a) FIR filter described by the difference equation : y(n) = x (n) + x (n – 4) (i) Compute and sketch magnitude and phase response. [4] ⎛π ⎞ ⎛π ⎞ (ii) Find its response to the input x(n) = cos⎜ n ⎟ + cos⎜ n ⎟, − ∞ < n < ∞. ⎝2 ⎠ ⎝4 ⎠ [4] Solution : (i) To find Magnitude and Phase Response Given (i) By ZT, y (n) = x (n) + x (n – 4) Y (z) = x (z) + z -4 x (z = x (z) (1 + z -4) H (z) = 1 + z -4 z = e jw H (e jw) = 1 + e –j4w Put = e − j2 w e j2 w + e − j2 w H (e jw ) = e − j2 w [2 cos (2w )] (i) (ii) (iii) w 0 0.1 π 0.2 π 0.3 π 0.4 π 0.5 π 0.6 π 0.7 π 0.8 π 0.9 π π [ ] Magnitude Response M (w) = | Hr(w) | = | 2 cos (2w) | Phase Response : φ ( w ) = e − j2 w Phase …show more content…

y (n) = x (n) * h (n) To find Linear convolution using circular convolution, I. Select N=L+M–1=4+3–1=6 II. Append x[n] by (N – L = 2) zeros and h (n) by (N – m = 3) zeros x (n) = { 1, 2, 3, 4, 0, 0 } ∴ h (n) = { 2, 3, 1, 0, 0, 0 } III. Find y(n) using circular convolution N −1 y( n ) = x ( n ) ⊗ h ( n ) = y(n ) = 5 m =0 ∑ x ( m ) h ( n − m) where N = 6 m =0 ∑ x ( m ) h ( n − m) 5 i) n = 0, y (0) = m=0 ∑ x ( m) h ( − m) 2 y(0) = (1)( 2) + ( 2)(0) + (3)(0) + (4)(0) + (0)(1) + (0)(3) = ii) n = 1, y(1) = y(1) = (1)(3) + ( 2)( 2) + (3)(0) + ( 4)(0) + (0) + (0)(1) = iii) n = 2, y(2) = m =0 ∑ x(m) h (1 − m) 7 5 5 y(2) = (1)(1) + ( 2)(3) + (3)( 2) + 0 + 0 + 0 = 13 DSP Help Line : 9987030881 m =0 ∑ x ( m) h ( 2 − m ) www.guideforengineers.com B E EXTC iv) n = 3, y(3) = 5 DTS P DEC- 2004 6 y(3) = 0 + ( 2)(1) + (3)(3) + (4)( 2) + 0 + 0 = 19 v) n = 4, y(4) = m =0 ∑ x(m) h (3 − m) 5 y(4) = 0 + 0 + (3)(1) + ( 4)(3) + (0) + (0) = 15 m =0 ∑ x ( m) h ( 4 − m) 5 vi) n = 5, y(5) = = 0 + 0 + 0 + ( 4)(1) + 0 + 0 = m =0 ∑ x(m) h(5 − m) 4 ANS y[n] = { 2 , 7, 13, 19, 15, 4 } ↑ --------------------------------------------------------------------------------------------------------------Q 2. (a) The real sequence of length 8 is given as x[n] = {1, 2, 2, 0, 1, 1, 1 } Find the 8 point DFT X[k], by using 4 point DFTs only. Prove the property which is used. Solution (a) To find X[k] Given