Signal Processing and Linear Systems-B.P.Lathi copy

This function is i t 1 its mathematical description

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Unformatted text preview: <O Let i d(t) represent the function i tt) delayed (right-shifted) by 1 second as illustrated in Fig. 1.9b. This function is i (t - 1); its mathematical description can be obtained from i tt) by replacing t with t - 1 in Eq. (1.10). Thus ! d(t) = i (t - 1) = procedure if the signal is advanced (left-shifted) by 1 second. Show that this advanced signal fa(t) can be described as fa(t) = 2(t + 1) for - 1 ::; t ::; 0, and equal to 0 otherwise. \ l 4>(~) = f (t) (1.13) 4>(t) = f (2t) (1.14) a nd Observe t hat b ecause f (t) = 0 a t t = Tl a nd T2, we m ust h ave 4>(t) = 0 a t t = T d2 a nd T2/2, as s hown in Fig. 1.10b. I f f (t) were recorded o n a t ape a nd p layed back a t twice t he n ormal r ecording speed, we would o btain f (2t). I n general, if f (t) is compressed in t ime b y a factor a (a > 1), t he r esulting signal 4>(t) is given by 4>(t) = f (at) (1.15) Using a similar a rgument, we c an show t hat f (t) e xpanded (slowed down) in t ime b y a factor a (a > 1) is given by 4>(t) = f m (1.16) F igure 1.lOc shows f(~), which is f (t) e xpanded i n t ime b y a factor of 2. O bserve t hat i n t ime scaling operation, t he origin t = 0 is t he a nchor point, which remains unchanged u nder s caling o peration b ecause a t t = 0, f (t) = f (at) = f(O). I n s ummary, t o t ime-scale a signal by a factor a, we r eplace t w ith at. I f a > 1, t he s caling results in compression, a nd if a < 1, t he s caling results in expansion. 64 1 I ntroduction t o S ignals a nd S ystems 2 1.3 65 S ome U seful S ignal O perations f (t) (a) -2 5 1 --- - 1.5 o 1- 'P(t) =f (-I) (b) 2 t --- -5 -\ (b) 1- F ig. 1 .12 T ime inversion (reflection) of a signal. ...-----2· (e) - 1.5 :::; ~ < 0 or O:::;~<3 or otherwise -3 o 1- 6 O :::;t<6 Observe t hat t he i nstants t = - 1.5 a nd 3 in f (t) c orrespond t o t he i nstants t in t he e xpanded signal f(~). • (a) signal f (t) (b) signal f (3t) (c) signal f(~). F ig. 1 .11 - 3:::; t < 0 (1.18b) = - 3 a nd 6 ;:, E xercise E 1.6 • E xample 1 .4 F igure 1.11a shows a signal f (t). Sketch a nd describe mathematically this signal time-compressed b y factor 3. Repeat t he problem for t he same signal time-expanded by factor 2. T he signal f (t) c an be described as f (t) = {~e-t/2 - 1.5:::; t < 0 O:::;t<3 (1.17) otherwise F igure 1 .l1b s hows fc(t), which is f (t) time-compressed by factor 3; consequently, it can b e described mathematically as f (3t), which is o btained by replacing t with 3t in t he r ight-hand side o f E q. 1.17. T hus fc(t) = f (3t) = { ~ - 1.5 :::; 3 t < 0 or 3t 2 e- / o :::; 3t < 3 Show t hat the time-compression by a factor n ( n > 1) of a sinusoid results in a sinusoid of the same amplitude and phase, but with the frequency increased n-fold. Similarly the time expansion by a factor n ( n > 1) of a sinusoid results in a sinusoid of the same amplitude and phase, but with the frequency reduced by a factor n . Verify your conclusion by sketching a sinusoid sin 2 t...
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