Suppose x(t) = sin 2007Tt + 2 sin 4007Tt and g(t) = x(t) sin 4007Tt. If the product

Let x(t) be a signal for which X(jw) = 0 when lwl > WM. Another signal y(t) is specified as having the Fourier transform Y(jw) = 2X(j(w - we)). Determine a signal m(t) such that x(t) = y(t)m(t).

Let x(t) be a real-valued signal for which X(jw) = 0 when lwl > 1,0007T. Supposing that y(t) = ejwct x(t), answer the following questions: (a) What constraint should be placed on We to ensure that x(t) is recoverable from y(t)? (b) What constraint should be placed on We to ensure that x(t) is recoverable from (Re{y(t)}?

Let \(x(t)\) be a real-valued signal for which \(X(j \omega)=0\) when \(|\omega|>2,000 \pi\). Amplitude modulation is performed to produce the signal \(g(t)=x(t) \sin (2,000 \pi t)\). A proposed demodulation technique is illustrated in Figure P8.3 where \(g(t)\) is the input, \(y(t)\) is the output, and the ideal lowpass filter has cutoff frequency \(2,000 \pi\) and passband gain of 2. Determine \(y(t)\). Figure P8.3

Suppose x(t) = sin 200\(\pi\)t + 2 sin 400\(\pi\)t and g(t) = x(t) sin 400\(\pi\)t. If the product g(t)(sin 400\(\pi\)t) is passed through an ideal low pass filter with cutoff frequency 400\(\pi\) and passband gain of 2, determine the signal obtained at the output of the lowpass filter.

Suppose we wish to transmit the signal ( ) sin 1,0007Tt xt = 7Tt using a modulator that creates the signal w(t) = (x(t) + A) cos(l0,0007Tt). Determine the largest permissible value of the modulation index m that would allow asynchronous demodulation to be used to recover x(t) from w(t). For this problem, you should assume that the maximum magnitude taken on by a side lobe of a sine function occurs at the instant of time that is exactly halfway between the two zerocrossings enclosing the side lobe.

Assume that x(t) is a signal whose Fourier transform X(jw) is zero for lwl > WM. The signal g(t) may be expressed in terms of x(t) as where* denotes convolution and We> WM. Determine the value of the constant A such that x(t) = (g(t) cos Wet)* . 1Tt

An AM-SSB/SC system is applied to a signal x(t) whose Fourier transform X(jw) is zero for lw I > w M. The carrier frequency w c used in the system is greater than w M. Let g(t) denote the output of the system, assuming that only the upper sidebands are retained. Let q(t) denote the output of the system, assuming that only the lower sidebands are retained. The system in Figure P8.7 is proposed for converting g(t) into q(t). How should the parameter w0 in the figure be related to we? What should be the value of passband gain A?

Consider the modulation system shown in Figure P8.8. The input signal x(t) has a Fourier transform X(jw) that is zero for lw I > w M. Assuming that We > w M, answer the following questions: x(t) (a) Is y(t) guaranteed to be real if x(t) is real? (b) Can x(t) be recovered from y(t)?

Two signals x1 (t) and x2(t), each with a Fourier transform that is zero for lwl > we, are to be combined using frequency-division multiplexing. The AM-SSB/SC technique of Figure 8.21 is applied to each signal in a manner that retains the lower sidebands. The carrier frequencies used for x 1 (t) and x2(t) are We and 2we, respectively. The two modulated signals are then summed together to obtain the FDM signal y(t). (a) For what values of w is Y(jw) guaranteed to be zero? (b) Specify the values of A and w 0 so that [{ sin wot} l A sin Wet XJ (t) = y(t) * ~ COS Wot * Trt , where* denotes convolution.

Let X c(t) = L akejkw,r, k=-X where a0 = 0 and a 1 =rf 0, be a real-valued periodic signal. Also, let x(t) be a signal with X(jw) = 0 for lwl 2: wJ2. The signal x(t) is used to modulate the carrier c(t) to obtain y(t) = x(t)c(t). (a) Specify the passband and the passband gain of an ideal bandpass filter so that, with input y(t), the output of the filter is g(t) = (a I ejwct +a~ e-jw,.t)x(t). (b) If a1 = latlej

Consider a set of 10 signals xi(t), i = 1, 2, 3, ... , 10. Assume that each Xi(t) has Fourier transform such that Xi(jw) = 0 for lwl 2: 2,00071'. All10 signals are to be time-division multiplexed after each is multiplied by a carrier c(t) shown in Figure P8.12. If the period T of c(t) is chosen to have the maximum allowable value, what is the largest value of~ such that all 10 signals can be time-division multiplexed?

A class of popularly used pulses in PAM are those which have a raised cosine frequency response. The frequency response of one of the members of this class is P(jw) = { i(1 +coswJ'), 0::::; lwl::::; ~~, 0, elsewhere where T1 is the intersymbol spacing. (a) Determine p(O). (b) Determine p(kTI), where k = 1, 2, ....

Consider the frequency-modulated signal y(t) = cos( wet + m cos Wmt), where we>> Wm and m << 7T/2. Specify an approximation to Y(jw) for w > 0.

For what values of w 0 in the range -7r < w 0 ::::; 7T is amplitude modulation with carrier ejwon equivalent to amplitude modulation with carrier cos w0 n?

Suppose x[n] is a real-valued discrete-time signal whose Fourier transform X(ejw) has the property that . 7T X(elw) = 0 for S ::::; w ::::; 'TT. We use x[n] to modulate a sinusoidal carrier c[n] = sin(57T/2)n to produce y[n] = x[n]c[n]. Determine the values of win the range 0 ::::; w ::::; 7T for which Y(ejw) is guaranteed to be zero.

Consider an arbitrary finite-duration signal x[n] with Fourier transform X(ejw). We generate a signal g[n] through insertion of zero-valued samples: [ ] [ ] { x[n/4], n = 0, 4, 8, 12, ... g n = X(4) n = 0, otherwise The signal g[n] is passed through an ideallowpass filter with cutoff frequency 7T/4 and passband gain of unity to produce a signal q[n]. Finally, we obtain y[n] = q[n] cos e: n). For what values of w is Y(ejw) guaranteed to be zero?

Let x[n] be a real-valued discrete-time signal whose Fourier transform X(ejw) is zero for w 2: 7T/4. We wish to obtain a signal y[n] whose Fourier transform has the property that, in the interval -7r < w :::; 1T, { X(ej(w-~)), Y(ejw) = X(ej(w+~)), 0, '!!_ < w < 37T 2 - 4 - 37T < w :::; 4 otherwise 7T 2 The system in Figure P8.18 is proposed for obtaining y[n] from x[n]. Determine constraints that the frequency response H ( ejw) of the filter in the figure must satisfy for the proposed system to work.

Consider 10 arbitrary real-valued signals Xi[n], i = 1, 2, ... , 10. Suppose each Xi[n] is upsampled by a factor of N, and then sinusoidal amplitude modulation is applied to it with carrier frequency wi = i1TI10. Determine the value of N which would guarantee that all 10 modulated signals can be summed together to yield an FDM signal y[n] from which each Xi[n] can be recovered.

In Sections 8.1 and 8.2, we analyzed the sinusoidal amplitude modulation and demodulation system of Figure 8.8, assuming that the phase 8 c of the carrier signal was zero. 8.22. cos(5wt) (a) For the more general case of arbitrary phase 8 c in the figure, show that the signal in the demodulation system can be expressed as 1 1 w(t) = 2x(t) + 2x(t)cos(2wct + 28c). (b) If x(t) has a spectrum that is zero for lw I 2:: w M, determine the relationships required among w co [the cutoff frequency of the ideallowpass filter in Figure 8.8(b)], We (the carrier frequency), and WM so that the output of the lowpass filter is proportional to x(t). Does your answer depend on the carrier phase 8 c?

In Figure P8.22(a), a system is shown with input signal x(t) and output signal y(t). The input signal has the Fourier transform X(jw) shown in Figure P8.22(b). Determine and sketch Y(jw ), the spectrum of y(t).

In Section 8.2, we discussed the effect of a loss of synchronization in phase between the carrier signals in the modulator and demodulator in sinusoidal amplitude modulation. We showed that the output of the demodulation is attenuated by the cosine of the phase difference, and in particular, when the modulator and demodulator have a phase difference of Tr/2, the demodulator output is zero. As we demonstrate in this problem, it is also important to have frequency synchronization between the modulator and demodulator. Consider the amplitude modulation and demodulation systems in Figure 8.8 with () c = 0 and with a change in the frequency of the demodulator carrier so that w(t) = y(t) cos wc~t, where y(t) = x(t) cos Wet. Let us denote the difference in frequency between the modulator and demodulator as ~w (i.e., wd- We = ~w ). Also, assume that x(t) is band limited with X(jw) = 0 for lw I 2 w M, and assume that the cutoff frequency w co of the lowpass filter in the demodulator satisfies the inequality (a) Show that the output of the lowpass filter in the demodulator is proportional to x(t) cos(~wt). (b) If the spectrum of x(t) is that shown in Figure P8.23, sketch the spectrum of the output of the demodulator.

Figure P8.24 shows a system to be used for sinusoidal amplitude modulation, where x(t) is band limited with maximum frequency WM, so that X(jw) = 0, lwl > WM. As indicated, the signal s(t) is a periodic impulse train with period T and with an offset from t = 0 of d. The system H(jw) is a bandpass filter. (a) With Li = 0, WM = 7ri2T, w1 = 7r!T, and wh = 37r/T, show that y(t) is proportional to x(t) cos wet, where We = 27r!T. (b) If WM, w1, and wh are the same as given in part (a), but dis not necessarily zero, show that y(t) is proportional to x(t) cos(wct + Oc), and determine We and 0 c as a function of T and d. (c) Determine the maximum allowable value of WM relative toT such that y(t) is proportional to x(t) cos( wet + 0 c).

A commonly used system to maintain privacy in voice communication is a speech scrambler. As illustrated in Figure P8.25(a), the input to the system is a normal speech signal x(t) and the output is the scrambled version y(t). The signal y(t) is transmitted and then unscrambled at the receiver. We assume that all inputs to the scrambler are real and band limited to the frequency w M; that is, X (j w) = 0 for lw I > w M. Given any such input, our proposed scrambler permutes different bands of the spectrum of the input signal. In addition, the output signal is real and band limited to the same frequency band; that is, Y(jw) = 0 for lw I > w M. The specific algorithm for the scrambler is Y(jw) = X(j(w- WM)), w > 0, Y(jw) = X(j(w + WM)), w < 0. (a) If X(jw) is given by the spectrum shown in Figure P8.25(b), sketch the spectrum of the scrambled signal y(t). (b) Using amplifiers, multipliers, adders, oscillators, and whatever ideal filters you find necessary, draw the block diagram for such an ideal scrambler. (c) Again using amplifiers, multipliers, adders, oscillators, and ideal filters, draw a block diagram for the associated unscrambler.

In Section 8.2.2, we discussed the use of an envelope detector for asynchronous demodulation of an AM signal of the form y(t) = [x(t) + A] cos(wct + Oc). An alternative demodulation system, which also does not require phase synchronization, but does require frequency synchronization, is shown in block diagram form in Figure P8.26. The lowpass filters both have a cutoff frequency of We. The signal y(t) = [x(t) +A] cos(wct + Oc), with Oc constant but unknown. The signal x(t) is band limited with X(jw) = 0, lwl > WM, and with WM 0 for all t. Show that the system in Figure P8.26 can be used to recover x(t) from y(t) without knowledge of the modulator phase e c

As discussed in Section 8.2.2, asynchronous modulation-demodulation requires the injection of the carrier signal so that the modulated signal is of the form y(t) = [A + x(t)] cos(wct + Oc), (P8.27-1) where A+ x(t) > 0 for all t. The presence of the carrier means that more transmitter power is required, representing an inefficiency. (a) Let x(t) = cos WMt with WM 0. For a periodic signal y(t) with period T, the average power over time is defined asPy = (1/T) JT y2(t) dt. Determine and sketch Py for y(t) in eq. (P8.27-1). Express your answer as a function of the modulation index m, defined as the maximum absolute value of x(t) divided by A. (b) The efficiency of transmission of an amplitude-modulated signal is defined to be the ratio of the power in the sidebands of the signal to the total power in the signal. With x(t) = coswMt, and with WM 0, determine and sketch the efficiency of the modulated signal as a function of the modulation index m.

In Section 8.4 we discussed the implementation of single-sideband modulation using 90 phase-shift networks, and in Figures 8.21 and 8.22 we specifically illustrated the system and associated spectra required to retain the lower sidebands. x(t) Figure P8.28(a) shows the corresponding system required to retain the upper sidebands. (a) With the same X(jw) illustrated in Figure 8.22, sketch Y1 (jw ), Y2(jw ), and Y(jw) for the system in Figure P8.28(a), and demonstrate that only the upper sidebands are retained. (b) For X(jw) imaginary, as illustrated in Figure P8.28(b), sketch Y 1(jw), Y2(jw), and Y(jw) for the system in Figure P8.28(a), and demonstrate that, for this case also, only the upper sidebands are retained.

Single-sideband modulation is commonly used in point-to-point voice communication. It offers many advantages, including effective use of available power, conservation of bandwidth, and insensitivity to some forms of random fading in the channel. In double-sideband suppressed carrier (DSB/SC) systems the spectrum of the modulating signal appears in its entirety in two places in the transmitted spectrum. Single-sideband modulation eliminates this redundancy, thus conserving bandwidth and increasing the signal-to-noise ratio within the remaining portion of the spectrum that is transmitted. In Figure P8.29(a), two systems for generating an amplitude-modulated single-sideband signal are shown. The system on the top can be used to generate a single-sideband signal for which the lower sideband is retained, and the system on the bottom can produce a single-sideband signal for which the upper sideband is retained. (a) For X(jw) as shown in Figure P8.29(b), determine and sketch S(jw), the Fourier transform of the lower sideband modulated signal, and R(jw ), the Fourier transform of the upper sideband modulated signal. Assume that We> W3. The upper sideband modulation scheme is particularly useful with voice communication, as any real filter has a finite transition region for the cutoff (i.e., near we). This region can be accommodated with negligible distortion, since the voice signal does not have any significant energy near w = 0 (i.e., for jwj < Wt = 27T X 40Hz). (b) Another procedure for generating a single-sideband signal is termed the phaseshift method and is illustrated in Figure P8.29(c). Show that the singlesideband signal generated is proportional to that generated by the lower sideband modulation scheme of Figure P8.29(a) [i.e., p(t) is proportional to s(t)]. (c) All three AM-SSB signals can be demodulated using the scheme shown on the right-hand side of Figure P8.29(a). Show that, whether the received signal is s(t), r(t), or p(t), as long as the oscillator at the receiver is in phase with oscillators at the transmitter, and w = We, the output of the demodulator is x(t). The distortion that results when the oscillator is not in phase with the transmitter, called quadrature distortion, can be particularly troublesome in data communication.

Let x[n] be a discrete-time signal with spectrum \(X\left(e^{j \omega}\right)\), and let p(t) be a continuous time pulse function with spectrum P(j\(\omega\)). We form the signal \(y(t)=\sum_{n=-\infty}^{+\infty} x[n] p(t-n) .\) (a) Determine the spectrum \(Y(j \omega)\) in terms of \(X\left(e^{j \omega}\right)\) and P(j\(\omega\)). (b) If \(p(t)=\left\{\begin{array}{ll}\cos 8 \pi t, & 0 \leq t \leq 1 \\0, & \text { elsewhere }\end{array},\right.\) determine \(P(j \omega) \text { and } Y(j \omega)\).

Consider a discrete-time signal x[n] with Fourier transform shown in Figure P8.32(a). The signal is amplitude modulated by a sinusoidal sequence, as indicated in Figure P8.32(b) (a) Determine and sketch Y(e.iw), the Fourier transform of y[n]. (b) A proposed demodulation system is shown in Figure P8.32(c). For what value of fJ c. w1p, and G will x[n] = x[n]? Are any restrictions on we and w1P necessary to guarantee that x[n] is recoverable from y[n]?

Let us consider the frequency-division multiplexing of discrete-time signals xi[n], i = 0, 1, 2, 3. Furthermore, each Xi[n] potentially occupies the entire frequency band ( -7r < w < 1r). The sinusoidal modulation of upsampled versions of each of these signals may be carried out by using either double-sideband techniques or single-sideband techniques. (a) Suppose each signal Xi[n] is appropriately upsampled and then modulated with cos[i(1TI4)n]. What is the minimum amount ofupsampling that must be carried out on each xi[n] in order to ensure that the spectrum of the FDM signal does not have any aliasing? (b) If the upsampling of each xi[n] is restricted to be by a factor of 4, how would you use single-sideband techniques to ensure that the FDM signal does not have any aliasing? Hint: See Problem 8.17.

In discussing amplitude modulation systems, modulation and demodulation were carried out through the use of a multiplier. Since multipliers are often difficult to implement, many practical systems use a nonlinear element. In this problem, we illustrate the basic concept. In Figure P8.34, we show one such nonlinear system for amplitude modulation. The system consists of squaring the sum of the modulating signal and the carrier and then bandpass filtering to obtain the amplitude-modulated signal. Assume that x(t) is band limited, so that X(jw) = 0, lwl > WM. Determine the bandpass filter parameters A, w 1, and wh such that y(t) is an amplitudemodulated version of x(t) [i.e., such that y(t) = x(t) cos wet]. Specify the necessary constraints, if any, on we and w M

The modulation -demodulation scheme proposed in this problem is similar to sinusoidal amplitude modulation, except that the demodulation is done with a square wave with the same zero-crossings as cos wet. The system is shown in Figure P8.35(a); the relation between cos wet and p(t) is shown in Figure P8.35(b). Let the input signal x(t) be a band-limited signal with maximum frequency WM

The accurate demultiplexing-demodulation of radio and television signals is generally performed using a system called the superheterodyne receiver, which is equivalent to a tunable filter. The basic system is shown in Figure P8.36(a). (a) The input signal y(t) consists of the superposition of many amplitude-modulated signals that have been multiplexed using frequency-division multiplexing, so that each signal occupies a different frequency channel. Let us consider one such channel that contains the amplitude-modulated signal y1 (t) = x1 (t) cos wet, with spectrum Y1 (jw) as depicted at the top of Figure P8.36(b). We want to demultiplex and demodulate y 1 (t) to recover the modulating signal x 1 (t), using the system of Figure P8.36(a). The coarse tunable filter has the spectrum H1 (jw) shown at the bottom ofFigureP8.36(b). Determine the spectrumZ(jw) of the input signal to the fixed selective filter H2(jw ). Sketch and label Z(jw) for w >0. (b) The fixed frequency-selective filter is a bandpass type centered around the fixed frequency w 1, as shown in Figure P8.36(c). We would l~ke the output of the filter with spectrum H2(jw) to be r(t) = XJ(t)cosw 1t. In terms of We and WM, what constraint must WT satisfy to guarantee that an undistorted spectrum of x 1 (t) is centered around w = w f? (c) What must G, a, and {3 be in Figure P8.36(c) so that r(t) = x1 (t) cos w 1t?

The following scheme has been proposed to perform amplitude modulation: The input signal x(t) is added to the carrier signal cos wet and then put through a nonlineardevice, so that the output z(t) is related to the input by z(t) = eY

In Figure P8.38(a), a communication system is shown that transmits a band-limited signal x(t) as periodic bursts of high-frequency energy. Assume that X(jw) = 0 for lwl > WM. Two possible choices, m1 (t) and m2(t), are considered for the modulating signal m(t). m1 (t) is a periodic train of sinusoidal pulses, each of duration D, as shown in Figure P8.38(b). That is, m1 (t) = L, p(t - kT), k= -oc where ( ) _ { COS Wet, ltl < (D/2) p t - 0, ltl > (D/2) . m2(t) is cos Wet periodically blanked or gated; that is, m2(t) = g(t) cos wet, where g(t) is as shown in Figure P8.38(b ). The following relationships between the parameters T, D, We, and WM are assumed: D> D' 27T T >2wM. Also, assume that [sin(x)]/x is negligible for x >> 1. Determine whether, for some choice of w1p, either m1 (t) or m2(t) will result in a demodulated signal x(t). For each case in which your answer is yes, determine an acceptable range for WJp

Suppose we wish to communicate one of two possible messages: message m0 or message m1 To do so, we will send a burst of one of two frequencies over a time interval of length T. Note that Tis independent of which message is being transmitted. For message m0 we will send cos w0t, and for message m1 we will send cosw1t. Thus, a burst b(t) will look as shown in Figure P8.39(a). Such a communication system is called frequency shift keying (FSK). When the burst of frequency b(t) is received, we wish to determine whether it represents message m0 or message m1 To accomplish this, we do as illustrated in Figure P8.39(b). (a) Show that the maximum difference between the absolute values of the two lines in Figure P8.39(b) occurs when cos w 0 t and cos w 1 t have the relationship i,T cos wotcos w 1 t dt = 0. (b) Is it possible to choose w 0 and w 1 such that there is no interval of length T for which LT cosw0tcosw,tdt = 0?

In Problem 8.40, we introduced the concept of quadrature multiplexing, whereby two signals are summed after each has been modulated with carrier signals of identical frequency, but with a phase difference of 90. The corresponding discrete-time multiplexer and demultiplexer are shown in Figure P8.41. The signals x 1 [n] and x2 [n] are both assumed to be band limited with maximum frequency WM, so that (a) Determine the range of values for We so that x 1 [n] and x2[n] can be recovered from r[n]. (b) With We satisfying the conditions in part (a), determine H(efw) so that YI [n] = XI [n] and y2[n] = x2[n].

In order to avoid intersymbol interference, pulses used in PAM systems are designed to be zero at integer multiples of the symbol spacing TI. In this problem, we develop a class of pulses which are zero at t = kTI, k = 1, 2, 3, .... Consider a pulse PI (t) that is real and even and that has a Fourier transform PI (jw ). Also, assume that P1 (-jw + j ~ ) = - P1 ~w + j ~ } 0 <; w -; (a) Define a periodic sequence PI (t) with Fourier transform and show that P1(jw) = m~oo P1 (jw- jm ~7} P- (. ) p- ( .27T) I }W = - I ]W - } T; (b) Use the result of the previous part to show that for some T PI (t) = 0, t = kT, k = 0, 2, 4, .... (c) Use the result of the previous part to show that Pt(kTt) = 0, k = 1, 2, 3, .... (d) Show that a pulse p(t) with Fourier transform { 1 + PI(jw), P(jw) = Pt(jw), 0, also has the property that lwl ~ ; f; ~ lwl ~ ~7 otherwise p(kTI) = 0, k = 1, 2, 3, ....

The impulse response of a channel used for PAM communication is specified by h(t) = 10,000e-I,OOOtu(t). It is assumed that the phase response of the channel is approximately linear in the bandwidth of the channel. A pulse that is received after passing through the channel is processed using an LTI systemS with impulse response g(t) in order to compensate for the nonuniform gain over the channel bandwidth. (a) Verify that if g(t) has the Fourier transform G(jw) = A+ }Bw, where A and Bare real constants, then g(t) can compensate for the nonuniform gain over the channel bandwidth. Determine the values of A and B. (b) It is proposed that S be implemented with the system shown in Figure P8.43. Determine the values of the gain factors a and f3 in this system. x(t) (Received signal before compensation) x(t) Figure P8.43 y(t) (Received signal after compensation)

In this problem, we explore an equalization method used to avoid intersymbol interference caused in PAM ~ystems by the channel having nonlinear phase over its bandwidth. When a PAM pulse with zero-crossings at integer multiples of the symbol spacing T 1 is passed through a channel with nonlinear phase, the received pulse may no longer have zero-crossings at times that are integer multiples ofT 1 Therefore, in order to avoid intersymbol interference, the received pulse is passed through a zeroforcing equalizer, which forces the pulse to have zero-crossings at integer multiples of T1 This equalizer generates a new pulse y(t) by summing up weighted and shifted versions of the received pulse x(t). The pulse y(t) is given by N y(t) = L GtX(t- lTJ), 1=-N where the a1 are all real and are chosen such that y(kT1) ~ { 6: k = 0 k = 1, 2, 3, ... , N. (P8.44-1) (a) Show that the equalizer is a filter and determine its impulse response. (b) To illustrate the selection of the weights a1, let us consider an example. If x(OTJ) = 0.0, x(-TJ) = 0.2, x(TJ) = -0.2, and x(kT1) = 0 for lkl > 1, determine the values of a0 , a 1, and a_ 1 such that y(TJ) = 0.

A band-limited signal x(t) is to be transmitted using narrowband FM techniques. That is, the modulation index m, as defined in Section 8.7, is much less than Tr/2. Before x(t) is transmitted to the modulator, it is processed so that X(jw )iw =O = 0 and lx(t)l < 1. This normalized x(t) is now used to angle-modulate a carrier to form the FM signal (a) Determine the instantaneous frequency w;. (b) Using eqs. (8.44) and (8.45), the narrowband assumption (m << Tr/2), and the preceding normalization conditions, show that (c) What is the relationship among the bandwidth of y(t), the bandwidth of x(t), and the carrier frequency We?

Consider the complex exponential function of time, s(t) = ei8(t)' (P8.46-1) where O(t) = w 0 t 2/2. Since the instantaneous frequency w; = d8/dt is also a function of time, the signal s(t) may be regarded as an FM signal. In particular, since the signal sweeps linearly through the frequency spectrum with time, it is often called a frequency "chirp" or "chirp signal." (a) Determine the instantaneous frequency. (b) Determine and sketch the magnitude and phase of the Fourier transform of the "chirp signal." To evaluate the Fourier transform integral, you may find it helpful to complete the square in the exponent in the integrand and to use the relation J +oc . 2 H 1T e 12 dz = -(1 + j). (c) Consider the system in Figure P8.46, in which s(t) is the "chirp signal" in eq. (P8.46-1). Show that y(t) = X(jw0 t), where X(jw) is the Fourier transform of x(t). (Note: The system in Figure P8.46 is referred to as the "chirp" transform algorithm and is often used in practice to obtain the Fourier transform of a signal.)

In Section 8.8 we considered synchronous discrete-time modulation and demodulation with a sinusoidal carrier. In this problem we want to consider the effect of a loss in synchronization in phase and/or frequency. The modulation and demodulation systems are shown in Figure P8.47(a), where both a phase and frequency difference between the modulator and demodulator carriers is indicated. Let the frequency difference w d - w c be denoted as ~w and the phase difference () d - () c as ~e. (a) If the spectrum of x[n] is that shown in Figure P8.47(b), sketch the spectrum of w[n], assuming ~w = 0. (b) If ~w = 0, show that w can be chosen so that the output r[ n] is r[ n] = x[n] cos~(). In particular, what is r[n] if~() = n/2? (c) For~() = 0, and w = WM + ~w, show that the output r[n] = x[n] cos[~wn] (assume that ~w is small).

In this problem, we consider the analysis of discrete-time amplitude modulation of a pulse-train carrier. The system to be considered is shown in Figure P8.48(a). (a) Determine and sketch the discrete-time Fourier transform of the periodic square-wave signal p[n] in Figure P8.48(a). (b) Assume that x[ n] has the spectrum shown in Figure P8.48(b ). With wM = 7ri2N and with M = 1 in Figure P8.48(a), sketch Y(ejw), the Fourier transform of y[n]. (c) Now assume that X(ejw) is known to be band limited with X(ejw) = 0, WM < w < 27r- WM, but is otherwise unspecified. For the system of Figure P8.48(a), determine, as a function of N, the maximum allowable value of w M that will permit x[n] to be recovered from y[n]. Indicate whether your result depends onM. (d) With w M and N satisfying the condition determined in part (c), state or show in block diagram form how to recover x[n] from y[n].

In practice it is often very difficult to build an amplifier at very low frequencies. Consequently~ low-frequency amplifiers typically exploit the principles of amplitude modulation to shift the signal into a higher-frequency band. Such an amplifier is referred to as a chopper amplifier and is illustrated in the block-diagram form in Figure P8.49. (a) Determine in terms ofT the highest allowable frequency present in x(t), if y(t) is to be proportional to x(t) (i.e., if the overall system is to be equivalent to an amplifier). (b) With x(t) bandlimited as specified in part (a), determine the gain of the overall system in Figure P8.49 in terms of A and T.