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Unformatted text preview: ign a lowpass Chebyshev filter to satisfy t he specifications:
- 25 d B, W p = 10 r ad/s, a nd W s = 15 r ad/s. 7 .6-4 Design a lowpass Chebyshev filter whose 3 dB cutoff frequency is
drops to - 50 d B a t 3we. 7 .7-1 F ind t he trll:nsfer f unction ! f(s) for a highpass B utterworth filter to satisfy t he specifications: G s:S; - 20 d B, G p ~ - 1 dB, w . = 10, a nd w p = 20. 7 .7-2 F ind t he tr,ansfer function. H (s) for a highpass Chebyshev filter t o s atisfy t he specifications: G s:S; - 22 dB, G p ~ - 1 d B, w . = 10, a nd W p = 20 7 .7-3 F ind t he t ra?sfer f unction HSs) for a B utterworth b andpass filter t o s atisfy t he specifications: G .:S; - 17 dB, G p ~ - 3 dB, W Pl = 100 r ad/s, W P2 = 250 r ad/s, a nd
w" = 40 r ad/s, w S 2 = 500 r ad/s. 7 .7-4 F ind t he tr,ansfer f unction H (s) for a Chebyshev b andpass filter t o s atisfy t he specifications: G s:S; - 17 d B, f :s; 1 d B, w P1 = 100 r ad/s, w P2 = 250 r ad/s, a nd W S1 = 40
r ad/s, w ' 2 = 500 r ad/s. 7 .7-5 F ind t he tr~nsfer function.H(s) for a B utterworth b andstop filter to satisfy t he specifications: G s:S; - 24 dB, G p ~ - 3 d B, w P1 = 20 r ad/s, W P2 = 60 r ad/s, a nd W S 1 = 30
r ad/s, W S2 = 38 r ad/s. ( d) T he s ystem gain a t dc times its 3 dB b andwidth is called t he g ain-bandwidth
p roduct of a system. Show t hat t his p roduct is t he same for all t he t hree s ystems in
Fig. P 7.3-1. T his r esult shows t hat if we increase t he b andwidth, t he g ain decreases
a nd vice versa.
7 .4-1 Using t he g raphical m ethod o f Sec 7.4-1, draw a rough sketch of t he a mplitude and
phase r esponse o f a n LTIC s ystem described by t he t ransfer function
S2 - H(s) 2s + 50 (s - 1 - j7)(s - 1 + j 7) = S2 + 2s + 50 = (s + 1 - j7)(s + 1 + j7) W hat k ind o f filter is t his?
1m 1m -2 Re .... -1 -2 -1 Re .... (b) (a) F ig. P 7.4-2 a ~ - 1 d B, ap ~ - 2 dB, a s p W e, as :s;
:5 a nd t he g ain 8.2 541 Some Useful Discrete-time Signal models
f [kl or f (kT) k- -2T T 5T T- JOT Fig. 8 .1 A d iscrete-time signal. D iscrete-Time
Signals and S ystems
I n this c hapter we i ntroduce t he basic concepts of discrete-time signals a nd
systems. 8.1 Introduction
Signals specified over a continuous range of t are c ontinuous-time s ignals,
d enoted by t he symbols f (t), y(t), etc. Systems whose inputs a nd o utputs a re
continuous-time signals are c ontinuous-time s ystems. I n contrast, signals defined
only a t d iscrete instants of time are d iscrete-time s ignals. Systems whose inputs
a nd o utputs a re discrete-time signals are called d iscrete-time s ystems. A digital
c omputer is a familiar example of this type of system. We consider here uniformly
spaced discrete instants such as . .. , - 2T, - T, 0, T, 2T, 3T, . .. , kT, . ... Discretetime signals can therefore be specified as f (kT), y(kT), a nd so o n (k, integer).
We further simplify this notation t o f [k],y[k], etc., where i t is u nderstood t hat
f [k] = f (kT) a nd t hat k is a n integer. A typical discrete-time si...
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