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If the discrete-time system is an ideal low-pass filter with a cutoff frequency of n/4, find the Fourier transform of y o ( [ )when ( a ) f i = 20 kHz and f 2 = 10 kHz and ( b )f l = 10 kHz and fi = 20 kHz. For bandlimited inpul signals, the system shown in Fig. 3-10 is a linear time-invariant continuous-time system. If

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find the frequency response of the equivalent continuous-time system. For bandlimited input signals, the system shown in Fig. 3- 1 is a linear time-invariant continuous-time system. If 0 the overall system is to be a differentiator,

The up-sampler and down-sampler are components that are found in interpolators and decimators, respectively. Are these systems linear Are they shift-invariant A sequence x ( n ) corresponds to samples of a bandlimited signal using a sampling frequency of 10 kHz. However, the sequence should have been sampled using a sampling frequency f , = 12 kHz. Design a system for digitally changing the sampling rate.

A signal x,(r) that is bandlimited to 10 kHz is processed by the following system:

express the output ya(t)in terms o f the input xa(t).

- 19 -

x l ( t )= cos(12rn~t) andx2(t)= c o s ( 1 7 2 ~ n t ) . ( a )4R,. ( b )R,/3.( c ) R,. 500 Hz. ( a ) 16 kHz. ( b ) 18 kHz.

come is $400 Since $50 is consumed regardless of the income level, $320 of the $370 level of consumption is induced by disposable income Induced consumption is $400 when disposable income is $500, and $480 when disposable income is $600 Solved Problem 42 What will happen to consumption function C in Figure 4-2 when: a Consumers consider their job secure and therefore become more con dent about the future level of disposable income b Credit card issuers implement tighter credit standards and consumers are less able to buy goods and services on credit c Consumers expect the price level to increase 10 percent by year end Solution: a Consumers become more willing to consume their current disposable income Consumption function C in Figure 4-2 shifts upward to C , and consumption is greater for each level of disposable income b.

Ha(jR)=

3.41 3.42 3.43

Both are linear and shift-varying. Up-sample by L = 6, filter with a low-pass filter that has a cutoff frequency of o,. ~ / and a gain of 6, and = 6 down-sample by M = 5.

4

# 127.0.0.1 128.95.135.13 128.95.135.24 128.95.142.30 140.27.133.4

The z-transform is a useful tool in the analysis of discrete-time signals and systems and is the discrete-time counterpart of the Laplace transform for continuous-time signals and systems. The z-transform may be used to solve constant coefficient difference equations, evaluate the response of a linear time-invariant system to a given input, and design linear filters. In this chapter, we will look at the z-transform and examine how it may be used to solve a variety of different problems.

In Chap. 2, we saw that the discrete-time Fourier transform (DTFT) of a sequence .c(n) is equal to the sum

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