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midterm1_sample_soln

midterm1_sample_soln - www ECE 342 Communication Theory...

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ECE 342 Communication Theory Fall 2005, Homework 1 Prof. Tiffany J Li www: http://www.eecs.lehigh.edu/ jingli/teach email: [email protected] Given: Wednesday Sept 28 5:35-6:55pm In class, closed book, 1 letter-size sheet of notes. 1. (20 points) Determine whether the following statements are true or false. If false, pro- vide a correct statement or a justification. (a) ( T ) Any signal that is a function of time has a frequency-domain representation that can be obtained using Fourier Transform. (b) ( F ) The amplitude spectrum (obtained by the Fourier Transform) of a signal is even symmetric while the phase spectrum is odd symmetric. (This holds for real signals only.) (c) ( T ) The Fourier transform of a periodic signal consists of a sequence of impulses in frequency at multiples of the fundamental frequency of the periodic signal. (d) ( T ) If x ( t ) ⇐⇒ X ( f ) , and y ( t ) ⇐⇒ Y ( f ) , then x ( t ) y ( t ) ⇐⇒ X ( f ) * Y ( f ) (i.e. multiplication in time translates to convolution in frequency). (e) ( F ) In a DSB-SC signal, the envelope of the resulting bandpass signal is pro- portional to the amplitude of the message signal. (The envelope of the resulting bandpass signal is proportional to the absolute value of the message signal’s amplitude.) 2. (10 points) Find the trigonometric Fourier series and sketch the corresponding spectra for the periodic impulse train g ( t ) = n = -∞ δ ( t - nT 0 ) . Solution: f 0 = 1 /T 0 and w 0 = 2 π/T 0 . The trigonometric Fourier series for g ( t ) is given by g ( t ) = a 0 + X n =1 ( a n cos( nw 0 t ) + b n sin( nw 0 t )) .

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midterm1_sample_soln - www ECE 342 Communication Theory...

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