102_1_Week-9-Recap - 38 NHAN LEVAN WEEK 9 RECAP 25 FT and...

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38 NHAN LEVAN WEEK 9 RECAP 25. FT and Linear TI Systems Q A & PROBLEMS FEST Q1. What is the meaning of Property-6 — Frequency-Shifting: e ± i ω 0 t f ( t ) | F ( i [ ω ω 0 ] ) ? A1. Consider the “modulated” signal f m ( t ) defined by f m ( t ) := f ( t ) cos ω 0 t, t R where f ( t ) := modulating signal, cos ω 0 t := carrier signal Let F m ( i ω ) denote the FT of f m ( t ) and F ( i ω ) is that of the f ( t ). Then F m ( i ω ) := F{ f ( t ) cos ω 0 t } = 1 2 [ F ( i [ ω - ω 0 ]) + F ( i [ ω + ω 0 ])] Thus the amplitude spectrum of f ( t ) is shifted by ± ω 0 . The modulating signal modulates the amplitude of cos ω 0 t . Hence the name amplitude modulation. P1. Show that for a LTI system S : x ( t ) [ S : LTI, h ( t )] y ( t ) , t R You have (25.1) Y ( i ω ) = H ( i ω ) X ( i ω ) , ω R S1 . By BT: (25.2) y ( t ) = -∞ h ( t - τ ) x ( τ ) d τ , t R This is also called a Convolution Integral. Taking the FT on both sides we get Y ( i ω ) = -∞ e - i ω t { -∞ h ( t - τ ) x ( τ ) d τ } dt, t R = -∞ x ( τ ) { -∞ e - i ω t h ( t - τ ) dt } d τ t - τ := σ Y ( i ω ) = -∞ x ( τ ) { -∞ e - i ω ( τ + σ ) h ( σ ) d σ } d τ Y ( i ω ) = -∞ e - i ωτ x ( τ ) { -∞ e - i ωσ h ( σ ) d σ } d τ = H ( i ω ) X ( i ω ) Conclusions-1: (i) Property 7: Time-Convolution F{ -∞ h ( t - τ ) x ( τ ) d τ } = H ( i ω ) X ( i ω )
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39 (ii) Application to IPOP of LTI Systems: t - Domain : x ( t ) [ S : LTI , h ( t )] y ( t ) = -∞ h ( t - τ ) x ( τ ) d τ , t R f - Domain : X ( i ω ) [ S : LTI , H ( i ω )] Y ( i ω ) = H ( i ω ) X ( i ω ) , ω R (iii) Definitions of FRF: H ( i ω ) := Frequency Response Function (FRF) = F{ h ( t ) } (25.3) := Y ( i ω ) X ( i ω ) = F{ OP } F{ IP } (25.4) P2. Find y(t) given that: t R : e i ω 0 t [ S : LTI, h ( t )] y ( t ) =? for some ω 0 R . S2 . BT y ( t ) = -∞ h ( t - τ ) e i ω 0 τ d τ t - τ := σ y ( t ) = e i ω 0 t -∞ e - i ω 0 σ h ( σ ) d σ = H ( i ω ) | ω = ω 0 e i ω 0 t , t R Result: (25.5) t R : e ± i ω 0 t [ S : LTI , h ( t )] y ( t ) = H ( ± i ω 0 ) e ± i ω 0 t , t R Important Remarks: (i) The signal e i ω 0 t , t R , is called a Steady-State Signal (SSS) since it keeps on going for a long long time, i.e., forever, i.e., everlasting! Note that e i ω t U ( t ) is not a SSS. (ii) It follows from (25.6) y ( t ) = H ( i ω 0 ) e i ω 0 t , t R that the SSS e i ω 0 t is very “special” for a L, TI, system S ! Why so? It is because of the fact that e i ω 0 t enters S then it reappears at the OP terminal as “itself” — except for the multiplicative constant H ( i ω 0 )! Such a function (signal) is called an Eigenfunction (Eigensignal) — of S . Moreover, it is clear that the corresponding OP y ( t ) is also a SSS. (iii) Returning to (25.6): y ( t ) = H ( i ω 0 ) e i ω 0 t , t R we have H ( i ω 0 ) := y ( t ) , t R e i ω 0 t , t R = OP due to e i ω 0 t , t R e i ω 0 t , t R So far the frequency ω 0 is arbitrary, that is, any ω 0 R . Hence one can drop the subscript 0. In so doing one gets the third definition for a FRF — besides (25.3) and (25.4).
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40 NHAN LEVAN Definition 4. The three faces of a FRF: H ( i ω ) := FRF = F{ h ( t ) } (25.7)
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