hw2 - ECE425 Fall 2006 Homework Set2 Your solutions to...

Unformatted text preview: ECE425. Fall 2006 Homework Set2 Your solutions to these problems are due in the ECE425 hand~in box at 5:00 PM on - Monday, Sept. 11, 2006. The hand-in box is located outside the south entrance to Phillips Hall 219 (usual place) and we use the upper right hand box. No late homework will be accepted without an official university excuse. Please work on these problems well in advance of the due date as we fully expect you may have questions and/or need suggestions. I Problem 4 Total Harmonic Distortion of a Stepped Approximation to a Sinewave The Total Harmonic Distortion (TH D) of a sinewave approximation is defined as: ...__ [A22 +A32 +A42 + u H H “11/2lA1 Where A1 is amplitude of the fundamental ("first harmonic"), A2 is the amplitude of the second harmonic, A3 the amplitude of the third harmonics, and so on. Show that THD of a stepped approximation-to a sinewave of exactly N steps per cycle is given to a good approximation, for large N, by: THD=er5N [One way to solve this problem would be to use a Fourier series to determine the Ak. This in itself would be a tedious calculation, which would depend on N of course. ln_this problem we will see how Using a sampling theory approach makes the problem much simpler] - Here are a serieson useful observations for you to consider in solving this problem. It is not necessary to study these observations in any particular order: (1) If the frequency of the sinewave is f, and we have exactly N samples in one cycle of the sinewave, explain why the "sampling rate" is fs=Nf.. (2) The problem of sampling a sinewave is an ordinary sampling problem. For a given N, sketch the spectrum to see what harmonics are present? (3) By the definition of THD, note that any common faCtors in amplitudes cancels out top and bottom. Given that the spectral shaping of the original sampling is the sample/hold roll-off, what common factors do we expect in all frequency components that are actually present? (4) Is it true that 1/(mN-1I)2+ 1/(mN+1)2 e 2/(mN)2? (5) The series 1 + 1/4 + 1/9 + 1/16 will likely appear. Look up this series sum if you can find it You can also infer the exact expression for the sum by knowing the Correct answer given to the the problem - assuming everything else is correct. Then verify this sum using Matlab.. (6) Given that there is only a (pathetic!) 1/N reduction of THD, what do we learn by considering the components that are actually non-zero (which we get from sampling theory) that suggests that stepwise sinewave generation will be usequ in practice? Problem 5 Cascade of a Discrete-Time and a Continuous-Time SamplelHold. A time sequence x(n) consists of samples that are non-zero only for even values of n” The odd valued samples are all of value 0‘. The samples are separated by a sampling time T.‘ It is desired to convert this sequence to a continuous-time signal where the value of each of the even samples is held for a period of 2T“ That is, the even samples are held for a period T and then for an additional period of-T (replacing the odd values of 0)“ ' If we'do this with an ordinary D/A converter, samples would only be held for a period T‘. What we need is first a discrete length-2 hold. The input to the discrete hold is x(n) and the output, y(n) would be: y(n) = x(n) n even y(n) = x(n-1) n odd This length—2 hold isjust a simple length-2 rectangular IOW-pass.‘ Following this, the ordinary sample/hold of length T achieves the desired result: (a) What is the frequency shaping of the S/H, H-r(f), that holds for length T? (b) What would be the frequency shaping of a S/H, H2T(f), that holds fer 2T? (c) What is the spectral shaping (frequency response) of the length-2 discrete hold, H2(f)? (d) It must be true that H2T(f) = H2(f) HT(f). Show this. Problem 6 Calculating the Freguency Response from the Impulse Response Consider a time sequence h(n) that represents the impulse response of a filter. Suppose first that the impulse response is h1(n)=1 for n=0,1,2,...5.. Find the frequency response H1(e“*’) that corresponds to h1(n), writing the result as the sum of cosines. Recalling that the “transfer function” corresponding to an impulse response h(n) is given as the “z—transform” of h(n): 00 H(z) = Z h(n)z'” =_OO [Note that this is exactly the more general form of the DTFT for the entire z-plane and not just at z=elw, the unit circle] H(z) is a polynomial in 2'1, with the h(n) being the coefficients of the polynomial. ' ' Write out H1(z) and find. the zeros. Compute the magnitude of the'frequency response H1(ej‘*’). Show that the zeros of H1(ej°‘) correSpond exactly to the zeros of H(z) that are on the unit circle. Repeat these steps of mm = [1 2 3 4 5 6 5 4 3 2 1]. What is the relationship between mm) and h2(n)? How is this reflected in the zeros of the transfer functions and in the manner in which the frequency response magnitudes approach zero value? ...
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