freq_resp

# freq_resp - Revisiting the Frequency Response Function Do...

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Revisiting the Frequency Response Function Do not distribute these notes 118 Prof. R.T. M’Closkey, UCLA

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General perspective of a system’s frequency response The idea behind the frequency response function for a linear system is simple: a sinusoidal input eventually pro- duces a sinusoidal output at the same frequency but with different magnitude and phase . We are assuming the system is asymptotically stable so transients due to initial conditions exponentially decay to zero. Pictorially, the idea is u ( t ) = cos( ωt ) - System - y ( t ) “large" t -→ A cos( ωt + φ ) As the frequency ω is changed, the magnitude A and phase φ of the output will, in general, also change. There are several ways to compute the amplitude and phase relationships as a function of frequency. For causal systems described by the n th order ODE , y ( n ) + a 1 y ( n - 1) + ··· + a n - 1 ˙ y + a n y = b 1 u ( n - 1) + ··· + b n - 1 ˙ u + b n u, (50) the frequency response function was derived as a particular solution of the ODE when the input was of the form u ( t ) r = e jωt . In this case, we can immediately write a particular solution of the same form using the transfer function (assuming s = is not a pole of the transfer function), y p ( t ) r = b 1 s n - 1 + ··· + b n - 1 s + b n s n + a 1 s n - 1 + ··· + a n - 1 s + a n ± ± ± ± s = e jωt r = b 1 ( ) n - 1 + ··· + b n - 1 ( ) + b n ( ) n + a 1 ( ) n - 1 + ··· + a n - 1 ( ) + a n e jωt r = H ( ) e jωt = | H ( ) | cos( ωt + H ( )) , Do not distribute these notes 119 Prof. R.T. M’Closkey, UCLA
where H represents the transfer function and s = . The solution of the IVP includes the initial conditions which we “lump" into the homogeneous term denoted y h , y ( t ) = y p ( t ) + y h ( t ) = | H ( ) | cos( ωt + H ( )) + y h ( t ) , t 0 . As t → ∞ the homogeneous term decays to zero at an exponential rate (assuming an asymptotically stable system), thus y ( t ) “large" t -→ | H ( ) | cos( ωt + H ( )) , (51) where the term “large" depends upon the time constants of the system. If a linear system is more generally described by convolution we can compute the sinusoidal output as follows when u ( t ) = cos ωt , for all t , y ( t ) = Z -∞ h ( t - τ ) u ( τ ) = Z -∞ h ( t - τ ) 1 2 ( e jωτ + e - jωτ ) = 1 2 Z -∞ h ( t - τ ) e jωτ + 1 2 Z -∞ h ( t - τ ) e - jωτ = 1 2 Z -∞ h ( s ) e ( t - s ) ds + 1 2 Z -∞ h ( s ) e - ( t - s ) ds = 1 2 ±Z -∞ h ( s ) e - jωs ds ² | {z } χ e jωt + 1 2 ±Z -∞ h ( s ) e jωs ds ² e - jωt , where χ C is deﬁned as shown. Thus, y may be written as y ( t ) = 1 2 χe jωt + 1 2 χ * e - jωt = 1 2 ³ χe jωt + ( χe jωt ) * ´ = | χ | cos( ωt + χ ) , (52) Do not distribute these notes 120 Prof. R.T. M’Closkey, UCLA

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where () * denotes the complex conjugate, | χ | is the magnitude of χ and χ is the phase angle of χ . We can think of χ as being a function of frequency, and for any particular frequency it evaluates to a complex number χ ( ω ) = Z -∞ h ( t ) e - jωt dt. (53) There are some requirements on h in order for this integral to exist. For example, if Z -∞ | h ( t ) | dt, (54) exists, then (53) also exists. In fact, the existence of (54) can be taken as the deﬁnition of “asymptotic" stability for systems that are more generally described by convolution.
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freq_resp - Revisiting the Frequency Response Function Do...

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