EECE 340 Homework, Spring 2002

EECE 340- Signals and Systems Homework # 2 Problem # 1 Consider the instantaneous sampling of the following sinusoidal signal: FQ) = 2c0s(@yt) It is desired to produce a DSB-SC signal from /(t) by sampling this signal and then using an ideal BPF of bandwidth equal to 2@,,. The desired carrier frequency of the DSB-SC signal is 500KHz, which is considered much bigger than f,,, Determine the smallest sampling period that can be used to obtain the DSB-SC signal. Problem #2 Determine the spectrum F(«) of the signal given below with > being a constant phase. S(t) = Acos(@@t + #). Problem # 3 Consider the following DSB-SC signal generated by the transmitter of a communication system: s(t) = f(P)cos(a@gft) . The signal f(t) is the information signal with bandwidth equal to J/OKHz. Due to time delay caused by signal propagation between transmitter and receiver, the signal at the receiver input is given by 5(0) = f@)cos(wgt + 4). Assume that ¢ is independent of time (constant). Determine the minimum bandwidth, BW, of the band pass filter that needs to be used in the receiver to pick s,(t). Use the result of Problem # 11 and the correspondence between time domain multiplication and frequency domain convolution to obtain the spectrum of s,(#). Page | of 3 Problem #4 Consider the following signal: t f®O=4 rea — 704 } Let f(t) be present at the input of a LPF of bandwidth B Hz. The LPF output is sampled at the rate of (//T) samples/sec, Assume that the LPF is ideal and the sampling is instantaneous. Let B Hz be also the bandwidth of f(t) defined by the first zero-crossing of the spectrum of f(t) with the frequency axis. Determine the sampling rate that permits the LPF output signal to be retrieved from its sampled version by ideal low-pass filtering. Problem #5 Consider the following signal: t-10° t) = 2 rect . fO : x | Let f(t) be sampled at the rate of 50,000 samples/sec to obtain the discrete time signal f(a). Determine the magnitude of the Fourier transform of f(n); i.e., \F(@)h Problem # 6 Consider the periodicity of the Fourier transform, F(@), of a discrete time signal, ffn), and the frequency shift property of F(a); i.e. f(n)e/*"” <> F(a). Determine all the values of @ that make F(@-@)=F(a). Problem #7 Consider a DSB-SC signal p(t) given as follows: pP(t)= A, f(t)cos(@,t). The signal f(f) is of low-pass type and having no frequency components above BHz. The DSB-SC signal is present at the input of an ideal BPF having a center frequency a, bandwidth equal to 2B Hz and frequency response H(@). The height of the magnitude frequency response of the BPF is 1 and its phase frequency response is equal to zero, (a) Determine the output of the BPF. (b) Draw the block diagram of a system that can be used to demodulate the DSB-SC signal. Page 2 of 3 Problem # 8 Consider the following low-pass signal: fse* ult). The signal /(t) is sampled at a rate equal to //T , where T is the sampling period. Plot the magnitude spectrum of the sampled signal and determine the bandwidth of an ideal LPF that picks the frequency components in f(t) within its half-power bandwidth. Problem # 9 Let a discrete-time signal fn) be given by: f(n)=e°"" u(n)=a" u(n) The signal f(n) is obtained by sampling f(t) =@7 7 u(t )at t=nT. (a) Apply the Discrete-Time Fourier transform; i.e. F(@) = vf(n) el? to fn) n=—00 and determine F(a). (b) Determine and plot the magnitude spectrum of /(n); i.e., |F(a,]| , and say if this is consistent with the plot you obtained in Problem 8. Page 3 of 3 / EECE 34.0 — Homiwork #- 2 ReluJon ( £ onary 4 stems | Frebhm#é1 j : The ccpeetrim a the tnslartaa eres ly Sampled Visseey ff 4) a tren by 4 po Figg= =D F(w-4). 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