ece 313 hw 12 - ECE 313 Fall 2010 Homework 12 –...

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Unformatted text preview: ECE 313 Fall 2010 Homework 12 – Due on Thursday, Dec. 9 (All book problems are from Chapter 14  copies of the problems are attached.) 1. Problem 33, p. 570. 2. € Consider the CT signal xc ( t ) having CTFT X c ( f ) = 1 − 0.01 f , f < 100; X c ( f ) = 0, f ≥ 100. (a) Graph X c ( f ) . (b) Suppose t€ xc ( t ) is sampled with a sample rate equal to 1.5 × the Nyquist hat rate. Graph the DTFT X ( F ) of the sampled signal. (c) Now suppose that xc ( t ) is sampled with a sample rate equal to 0.5 × the € € Nyquist rate. Graph the DTFT X ( F ) of the sampled signal. € 3. 4. Problem 38, p. 570. 5. € Problem 37, €. 570. p Consider the CT signal xc ( t ) = sinc( 4 t ) . This signal is sampled with a sample ȹ n ȹ 1 period Ts = sec. to generate the DT signal x [ n ] = xc ȹ ȹ . ȹ 8 Ⱥ 8 € € € (a) Graph the CTFT X c ( f ) and DTFT X ( F ) . € (b) The DT signal x [ n ] is passed through a DT LTI filter having impulse response € € 6. € € ȹ 1 ȹ n h [ n ] = ȹ ȹ u[ n ] . Let y [ n ] denote the filter output. Graph the magnitude of its € € ȹ 2 Ⱥ DTFT – that is, graph Y ( F ) . € 1 (c) Now suppose that y [ n ] is input to an ideal interpolator with Ts = sec. to € 8 ∞ ȹ t − nTs ȹ € generate the reconstructed CT signal y r ( t ) = ∑ y [ n ] sincȹ ȹ . Graph ȹ Ts Ⱥ n =−∞ € € Yr ( f ) . Suppose that we want to build a differentiator system for CT signals that are € d bandlimited to B Hz. From the table on p. 370 of the textbook: xc ( t ) has CTFT dt j 2 π f X c ( f ) . So, we want a filter having frequency response H c ( f ) that is (approximately) proportional to j ⋅ f for f ≤ B . We do this by using the € following system, where the C/D block is an ideal sampler with sample period € TS (“C/D” stands for “continuous ­to ­discrete”) and the D/C block is an ideal € interpolator with sample period TS (“D/C” stands for “discrete ­to ­continuous”). € € € € 7. 1 The DT filter has frequency response H ( F ) = j sin(2πF ), F ≤ . 2 (a) Assuming that sin( x ) ≈ x for x ≤ 0.1π and that B = 15 kHz, what range of values can we use for TS ? 1 jx (b) Recalling that sin( x ) = {e € − e− jx } , find the impulse response of the DT filter. € € € 2j € A CT Hilbert Transformer is a −90 phase shifting network that is used in communication systems applications. That is, a Hilbert Transformer filter ideally € has frequency response H f = − j, f > 0; H H ( f ) = j, f < 0. € H( ) The output of a Hilbert Transformer filter is called the Hilbert Transform of the input. € (a) Graph the magnitude and phase responses of the ideal Hilbert Transformer filter. (b) Suppose that in our application we need to generate the Hilbert Transform of a CT signal xc ( t ) that is bandlimited to 10 kHz. We do this by sampling the signal € with sample period TS = 5 × 10−5 sec., putting the sampled signal through a DT filter, and putting the DT filter output through an ideal interpolator again having −5 TS € = 5 × 10 . Find the DT filter frequency response H ( F ) needed so that the interpolator output is the Hilbert Transform of xc ( t ) . € (c) Find the impulse response h [ n ] of the DT filter found in part (b). Make your answer as simple as possible (it should not be complex ­valued). € € Problems from Textbook € ...
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