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

This function is i t 1 its mathematical description

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online