Introduction to Digital Signal and System Analysis - Tutorsindia

More documents

Recommendations

Info

Introduction to Digital Signal and System Analysis Frequency Domain Analysis a k 1 3 0 = 2nexp − 4 n= 2πkn j 4 for 0 ≤ k ≤ 3 Therefore, a 1 = 4 [ 0 + 2× 1exp( 0) + 2× 2exp( 0) + 2× 3exp( 0) ] = ( 0 + 2 + 4 + 6) 3 0 = 1 4 , a 1 1 ⎡ ⎛ = ⎢0 + 2 × 1exp⎜ − 4 ⎣ ⎝ 1 = 4 = −1+ j 2p ⎞ ⎛ j ⎟ + 2 × 2exp⎜ − 4 ⎠ ⎝ ( 0 + 2( − j) + 4( −1) + 6( j) ) 2p 2 ⎞ ⎛ j ⎟ + 2 × 3exp⎜ − 4 ⎠ ⎝ 2p 3 ⎞⎤ j ⎟ 4 ⎥ ⎠⎦ a a 2 3 1 ⎡ ⎛ 2p 2 ⎞ ⎛ 2p 2 × 2 ⎞ ⎛ 2p 2 × 3 ⎞⎤ = 0 2 1exp 2 2exp 2 3exp 4 ⎢ + × ⎜ − j ⎟ + × ⎜− j ⎟ + × ⎜− j ⎟ 4 4 4 ⎥ ⎣ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎦ 1 = ( 0 + 2( −1) + 4(1) + 6( −1) ) 4 = −1 1 ⎡ ⎛ = ⎢0 + 2 × 1exp⎜ − 4 ⎣ ⎝ 1 = 4 = −1− j ( 0 + 2( j) + 4( −1) + 6( − j) ) 2p 3 ⎞ ⎛ 2p 3× 2 ⎞ ⎛ 2p 3× 3 ⎞⎤ j ⎟ + 2× 2exp⎜− j ⎟ + 2× 3exp⎜− j ⎟ 4 ⎥ ⎠ ⎝ 4 ⎠ ⎝ 4 ⎠⎦ The spectral coefficients are a k = [ 3, −1+ j, −1, −1− j] , and the modulus is a k = [3, 1.4142, 1, 1.4142] . Figure 4.2 illustrates one period of the signal and its spectral coefficients. 2π 4n Example 4.2 A sine wave x[ n] = sin .. With k=4 and N=64, the sine wave and its Fourier spectrum are shown 64 in the following Figure 4.3. It can be seen that the single sine wave gives a single spectral line at frequencies of 4 hz and 64-4=60 hz, or p π 15π and 2π − = 8 8 8 radians/sample. Download free ebooks at bookboon.com 44
Introduction to Digital Signal and System Analysis Frequency Domain Analysis 6 x=2n 4 2 0 0 0.5 1 1.5 2 2.5 3 n 3 Fourier Series 2 1 0 0 0.5 1 1.5 2 2.5 3 k Figure 4.2 Spectral coefficients of Fourier series . 1.5 x=sin(2*pi*4*n/64) 1 0.5 0 -0.5 -1 -1.5 0 10 20 30 40 50 60 tim e (sample) 1 Fourier Series 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 frequency Figure 4.3 Sine wave and its spectral coefficients of Fourier series 45 Download free ebooks at bookboon.com
Page 2 and 3: Weiji Wang Introduction to Digital
Page 4 and 5: Introduction to Digital Signal and
Page 94 and 95:
Introduction to Digital Signal and
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
show all

Introduction to Digital Signal and System Analysis - Tutorsindia

Create successful ePaper yourself

Delete template?

Save as template?