Signals and Systems 2nd Edition solutions

Signals and Systems 2
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).

Signals and Systems 2
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 recover

Signals and Systems 2
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

Signals and Systems 2
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 f

Signals and Systems 2
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 prob

Signals and Systems 2
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

Signals and Systems 2
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 syst

Signals and Systems 2
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)?

Signals and Systems 2
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

Signals and Systems 2
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 fil

Signals and Systems 2
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 ha

Signals and Systems 2
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

Signals and Systems 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.

Signals and Systems 2
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?

Signals and Systems 2
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 whi

Signals and Systems 2
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

Signals and Systems 2
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

Signals and Systems 2
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 summ

Signals and Systems 2
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 syste

Signals and Systems 2
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).

Signals and Systems 2
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

Signals and Systems 2
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 fil

Signals and Systems 2
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 assu

Signals and Systems 2
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 bloc

Signals and Systems 2
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 require

Signals and Systems 2
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 t

Signals and Systems 2
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

Signals and Systems 2
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

Signals and Systems 2
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 P8.3

Signals and Systems 2
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 doubl

Signals and Systems 2
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

Signals and Systems 2
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

Signals and Systems 2
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 am

Signals and Systems 2
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

Signals and Systems 2
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 sinusoida

Signals and Systems 2
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

Signals and Systems 2
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.

Signals and Systems 2
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 Fourie

Signals and Systems 2
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 sy

Signals and Systems 2
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 phas

Signals and Systems 2
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

Signals and Systems 2
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

Signals and Systems 2
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

Signals and Systems 2
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

Signals and Systems 2
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 illust