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1. There is a decrease in the demand for a commodity when the price of a substitute commodity increases. 2. When the supply curve is positively sloped, an increase in demand will result in a larger quantity supplied. 3. A surplus exists when the market price is above the equilibrium price. 4. Government subsidization of rms producing Good A results in an increase in the demand for Good A. 5. Demand is inelastic if the percentage increase in quantity exceeds the percentage decrease in price. 6. A decline in price leaves total revenue unchanged when ED = 1. Answers: 1. False; 2. True; 3. True; 4. False; 5. False; 6. True

(b) For this minimum sampling frequency, findthe interpolation formula for x,(t) in terms of x ( n ) .

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(a) Because the highest frequency in xa(t) is Q2, the Nyquist rate is 2Q2. However, note that if xa(t) is modulated

+ Q1)/2,

then y,(t) is a (complex) low-pass signal with a spectrum shown in the following figure:

Yu(jQ)

where Qo = (Q2 - Q1)/2. Thus, the Nyquist rate for yo(() is 2Q0 = Q2 - Q l , which suggests that x,(t) may be uniquely reconstructed from its samples x,(nT,) provided that

Summary

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If this condition is satisfied, xa(t) may be uniquely reconstructed from xs(t) using a bandpass filter with a frequency response as shown below.

(b) With a sampling frequency Q, = Q2 - Q l , the reconstruction filter is a complex bandpass filter with an impulse response

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SAMPLING Therefore, the output of the reconstruction filter, which produces the complex bandpass signal x,(t), is

Given a real-valued bandpass signal x , ( t ) with X a ( f ) = 0 for If l < f i and If 1 > f2, the Nyquist sampling theorem says that the minimum sampling frequency is fs = 2 f2. However, in some cases, the signal may be sampled at a lower rate.

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(a) Suppose that fl = 8 kHz and f2 = 10 kHz. Make a sketch of the discrete-time Fourier transform of x ( n ) = x,(nTs) if fs = I /T, = 4 kHz. ( 6 ) Define the bandwidth of the bandpass signal to be

Show that if f , > B / 2 and f2 is an integer multiple of the bandwidth B , no aliasing will occur if x a ( t ) is sampled at a sampling frequency fs = 2 8 .

which is formed by shifting X,(f ) by integer multiples of the sampling frequency and summing. With f, = 4 kHz, we have the spectrum sketched below.

Note that X,( f) is not aliased. Therefore, with the appropriate processing of x,(r), the signal x,(r) may be recovered from its samples. Finally, the DTFT of the discrete-time sequence x ( n ) = x,(nTs) is

x(c'")

- 210 -

(b) If fi is an integer multiple of B , we may express f i and f2 as follows:

Because X a ( f ) is nonzero only for (I - l ) B < If 1 < l B , there is only one term in the sum that contributes to X,( f ) in the frequency range O< f <B and only one term that contributes to the frequency range -B if < 0 (draw a picture as in part (a) to see this clearly). Therefore, there is no aliasing, and x,(t) may be sampled without aliasing if a sampling frequency f, = 2B.

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