Ch3Handouts_2

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Unformatted text preview: esponse of the moving average filter 51 52 x[n] = A cos(ωo n + φ), − ∞ < n < ∞ Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra Steady-State Response Steady-State Response • We can express the input x[n] as x[ n] = g[n] + g * [n] where g[n] = 1 Ae jφe jωo n • Because of linearity, the response v[n] to an input g[n] is given by • Now the output of the system for an input e jωo n is simply • Likewise, the output v*[n] to the input g*[n] is v * [n] = 1 Ae− jφ H (e − jωo )e − jωo n v[n] = 1 Ae jφ H (e jωo )e jωo n 2 2 H (e jωo )e jωo n 2 53 54 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra 9 Steady-State Response Steady-State Response • Combining the last two equations we get y[n] = v[n] + v * [n] • Thus, the output y[n] has the same sinusoidal waveform as the input with two differences: (1) the amplitude is multiplied by H (e jωo ) , the value of the magnitude function at ω = ωo (2) the output has a phase lag relative to the input by an amount θ(ωo ), the value of the phase function at ω = ωo = 1 Ae jφ H (e jωo )e jωo n + 1 Ae − jφ H (e − jωo )e − jωo n 2 2 = 1 A H (e jωo ) {e jθ ( ωo )e jφe jωo n + e − jθ( ωo )e − jφe − jωon } 2 = A H (e jωo ) cos(ωo n + θ(ωo) + φ) 55 56 Copyright © 2005, S. K. Mitra Response to a Causal Exponential Sequence 57 • The expression for the steady-state response developed earlier assumes that the system is initially relaxed before the app...
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This document was uploaded on 11/09/2013.

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