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Unformatted text preview: ECE 2704: Notes on frequency response Our goal in these notes is to derive the steady-state response of a stable linear time- invariant system to a signal with a single frequency. We show that if the input to a stable linear system with transfer function H ( s ) is x ( t ) = M cos( ωt + φ ) then the steady-state output is y ( t ) = M | H ( jω ) | cos( ωt + φ + ∠ H ( jω )) Let H ( s ) be a stable transfer function such that Y ( s ) = H ( s ) X ( s ). We begin with an equivalent input, although expressed in a different form. Suppose the input is x ( t ) = A cos( ωt ) + B sin( ωt ) , (1) then our question is the following: what is the output y ( t ) for t sufficiently large? In other words, we want to find the output y ( t ) after the transient response has died away. 1 The input signal First, a few notes on the input signal (1). If we choose M and φ such that A = M cos( φ ) B =- M sin( φ ) then the input signal can be written x ( t ) = M cos( ωt + φ ) (2) where M = √ A 2 + B 2 (3) φ =- tan- 1 ( B/A ) (4) To see this is the case, note that A 2 + B 2 = M 2 cos 2 φ + M 2 sin 2 φ = M 2 (cos 2 ( φ ) + sin 2 ( φ )) = M 2 and that B A =- sin( φ ) cos( φ ) implies φ =- tan-...
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- Fall '08
- Frequency, jω