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ECE111_2010SPRING_HW1_PROFSOLN_[0]

ECE111_2010SPRING_HW1_PROFSOLN_[0] - The Cooper Union...

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The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Problem Set I: Signals & Spectra SOLUTION KEY Spring 2010 1. Figure 1 shows a sawtooth waveform with period T and peak amplitude A . Compute its RMS value. Does the answer depend on T ? RMS value= h 1 T R T 0 x 2 ( t ) dt i 1 = 2 . We have x ( t ) = A T t . So: k x k RMS = ° 1 T Z T 0 A 2 T 2 t 2 dt ± 1 = 2 = ° A 2 T 3 1 3 T 3 ± 1 = 2 = A p 3 The answer does not depend on T , and is proportional to peak value A . Note the constant of proportionality is NOT 1 p 2 ; many people mistakenly associate the 1 p 2 factor with the de°nition of RMS value; that only applies for sinusoidal waveforms; as this example illustrates, 1 p 2 does not yield the RMS scaling factor for all waveforms! 2. Periodicity in discrete-time is a bit more complicated than periodicity in continuous- time. Speci°cally, a discrete-time signal x [ n ] is considered periodic only if x [ n + N ] = x [ n ] for some INTEGER N ; if the formula "appears±periodic for some non-integer value T , then the signal is not really periodic. In particular, that means discrete-time sinewaves may not be periodic. Another complication with periodicity is if (say in continuous-time) x has period T , then it also has period 2 T , 3 T , even ° T and so forth (i.e., if x ( t + T ) = x ( t ) , then x ( t + mT ) = x ( t ) for all integers m ); thus, let us agree that the period of a signal (whether continuous or discrete) is the smallest positive value for which periodicity holds. (a) In continuous-time, what is the period of e jt ? In discrete-time, is e jn periodic, and, if so, what is the period? Keep in mind that exp ( ) = 1 IF AND ONLY IF ° = 2 ±n for integer n ; period of exp ( jt ) is T = 2 ± ; on the other hand exp ( jn ) is not periodic in general if the continuous time period is not rational the corresponding discrete time signal is not periodic! (b) In continuous-time, what is the period of cos (3 ±t= 4) ? In discrete-time, is cos (3 ±n= 4) periodic and, if so, what is the period? T = 2 ±= (3 ±= 4) = 8 = 3 ; in the discrete time case, the period has to be an integer; think of the discrete period N necessarily being an integer multiple of T : this suggests N = 8 . In other words the discrete time signal repeats after three cycles of the underlying continuous-time waveform. Anyway to summarize: continuous time signal has period T = 8 = 3 sec , discrete-time signal has period N = 8 samples.
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