7-frequency_response_LTIsystems

7-frequency_response - SYSC5602 Digital Signal Processing FREQUENCY RESPONSE Defined only for stable systems Consider an LTI system with impulse

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SYSC5602: Digital Signal Processing Mohamed El-Tanany, Professor Department of Systems & Computer Engineering, Carleton University, Ottawa, Ontario 1 / 23 FREQUENCY RESPONSE Defined only for stable systems Consider an LTI system with impulse response h(n), and an input sequence of the form . The system output in the steady state is given by: Therefore, the phase/amplitude relations between the input and output sequences are described by the function: Therefore, the frequency response is given by the transfer function of the system, evaluated along the unit circle . If the input x(n) is a linear combination of complex exponentials: xn () e j ω 0 n = y n hn e j ω 0 n hm e j ω 0 nm m = e j ω 0 n e j ω 0 m m = == = He j ω 0 e j ω 0 m m = z m m = ze j ω 0 = Hz j ω 0 = =
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SYSC5602: Digital Signal Processing Mohamed El-Tanany, Professor Department of Systems & Computer Engineering, Carleton University, Ottawa, Ontario 2 / 23 Then the output, in the steady-state, is given by: Properties Of The Frequency Response: is periodic with period 2 π ; this means the magnitude has even symmetry, while the phase exhibits odd-symmetry. To show above is true: xn () C 1 e j ω 1 n C 2 e j ω 2 n …… C M e j ω M n ++ + = y n C 1 He j ω 1 e j ω 1 n C 2 j ω 2 e j ω 2 n ……… C M j ω M e j ω M n + = j ω j ω H e j ω = j ω hm e j ω m m = ω m cos m = jh m ω m sin m = == j ω e j ω m m = ω m cos m = m ω m sin m = + j ω [] = j ω ω m cos m = 2 ω m sin m = 2 + j ω
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SYSC5602: Digital Signal Processing Mohamed El-Tanany, Professor Department of Systems & Computer Engineering, Carleton University, Ottawa, Ontario 3 / 23 The frequency response can also be obtained using the FFT (will elaborate later) Example Consider the LTI system with transfer function a) Calculate, and plot the frequency response , magnitude and phase, as a function of frequency for the range . b) Assume that the input sequence is of the form: Calculate the output sequence in the steady state. Solution The frequency response may be calculated by hand as follows. He j ω () hm ω m sin m = ⎝⎠ ⎜⎟ ⎛⎞ ω m cos m = atan j ω == Hz z 4 z 2 + z 4 0.25 + --------------------- = j ω 0 ωπ ≤≤ xn 0.532 π n 4 ------ cos 1.125 π n 2 cos 0.17 π n π 4 -- + cos ++ =
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SYSC5602: Digital Signal Processing Mohamed El-Tanany, Professor Department of Systems & Computer Engineering, Carleton University, Ottawa, Ontario 4 / 23 The results are shown graphically for both the magnitude and phase functions. The figure shows conjugate symmetry (magnitude is even, phase is odd) around π /2 This He j ω () z 4 z 2 + z 4 0.25 + --------------------- ze j ω = e j 4 ω e j 2 ω + e j 4 ω 0.25 + ---------------------------- == j 0 e j 0 e j 0 + e j 0 0.25 + ----------------------- 2 1.25 ---------- 1.6 e j 0 = j π 4 -- ⎝⎠ ⎜⎟ ⎛⎞ e j π e j π 2 + e j π 0.25 + 1 j + 1 –0 . 2 5 + 1.8856 e j 0.7854 === j π 2 e j 2 π e j π + e j 2 π 0.25 + -------------------------- 11 10 . 2 5 + ------------------- 0 = j π e j 4 π e j 2 π + e j 4 π 0.25 + 2 1.25 1.6 e j 0 =
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SYSC5602: Digital Signal Processing Mohamed El-Tanany, Professor
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This note was uploaded on 10/24/2009 for the course SCE sysc5602 taught by Professor El-tanany during the Spring '09 term at Carleton CA.

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7-frequency_response - SYSC5602 Digital Signal Processing FREQUENCY RESPONSE Defined only for stable systems Consider an LTI system with impulse

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