Ee3801mid22024
.pdf
School
Columbia University**We aren't endorsed by this school
Course
ELEN 3801
Subject
Electrical Engineering
Date
Dec 17, 2024
Pages
2
Uploaded by DukeFreedomFinch21
ELEN E3801Fall 2024Midterm 2November 19, 2024INSTRUCTIONS:•Carry only a812”×11”formula sheet (two-sided) and a pencil or a pen withyou.The exam is closed-book, closed-note.No electronic device of anykind is allowed.•The duration of the exam is 75 minutes.•Submit the blue book only – do not forget to write down your name andUNI.1
Problem I[15 pts]Letf1(t) =−t[u(t+ 1)−u(t)],andf2(t) =u(t).Compute their convolution.Problem II[15 pts]Letf1(t) = sinπ3t−π4+ej2t,andf2(t) =e−tu(t).Compute their convolution.Problem III[20 pts]A real-valued periodic signal is given byf(t) =ae−j3t+j2e−jt+ 3 +bejt+ (2−j2)ej3t.(a) Findaandb, and the signal bandwidth in Hz.(b) Sketch the exponential Fourier spectra.(c) By inspection of the spectra in (b), sketch the trigonometric Fourier spectra.(d) Find the compact trigonometric Fourier series from these spectra.Problem IV[10 pts]Consider the pulse train signalδT0(t) =∑∞n=−∞δ(t−nT0).(a) Find its exponential Fourier series.(b) Suppose it is passed through an ideal bandpass filter with the passband [ωL, ωH],i.e.,|H(ω)|= 1 forωL≤ |ω| ≤ωH,and|H(ω)|= 0 otherwise;̸H(ω) =−ωt0.Find the outputsignaly(t).What property doesy(t) have?Problem V[10 pts]Find the inverse Fourier transform of the signalF(ω) = [u(ω+ 5)−u(ω+ 3)] + [u(ω−3)−u(ω−5)].Problem VI[15 pts]For the baseband signalm(t) = 2 cos 1000t+cos 2000t,and the DSB-SC signalm(t) cos 10,000t:(a) Sketch the spectrum of the DSB-SC signal, and identify the USB and LSB spectra;(b) Identify the frequencies in the baseband, and the corresponding frequencies in the DSC-SC, USB and LSB spectra;(c) Draw the diagram of the demodulator and specify the bandwidth of the lowpass filter.Problem VII[15 pts]A signalf(t) = sinc(200πt) is uniformly sampled at a rate of (i) 150 Hz, (ii) 200 Hz, (iii)300Hz. For each of the three cases:(a) Sketch the spectrum of the sampled signal;(b) Explain if you can recover the signalf(t) from the sampled signal;(c) If the sampled signal is passed through an ideal lowpass filter of bandwidth 100 Hz,sketch the spectrum of the output signal.2