1. Consider the following system to process continuous-time signals with discrete-time processing. Convert to Impulses: Rate.

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Lee Summers
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1 ] A I C Signals and Systems, Fall 2012 Practice Problems - Final xam (More come!) Remember also look at the Practice Problems from xams #1 and #2! 1. Consider the following system process continuous-time signals with discrete-time processing. Sampler: Freq = (a) Suppose that! #"%$'&)(+*,- / "%$, 456&7498:&<;=3333 Hz, and the discrete-time filter frequency response, which you recall is periodic with period.10, is given in >?@0BA%0DC as: /FG$H& FJI 0 I K FLKNMO0 Find the output PQ #"%$. (b) Suppose that! #"%$ has Fourier Transform RS /4$:&UTB /4VN;=3333$, 4W5X&Y498Z&[;=3333 Hz, and the discrete-time filter frequency response, which you recall is periodic with period.10, is given in >?@0BA%0DC as: /FG$\& ;1A ^ M_K FLK`M0 3 A otherwise Find the Fourier transform ab /4$ of the output Pc #"d$, and then use it find Pc #"d$. (c) Suppose that! #"%$e&7(+*,- / "d$, 456&f498:&Y;=3333 Hz, and the discrete-time filter is defined by the difference equation: Pc> gqcd&h3 ij29pc> gk?;lc mn!> gqc Find the output PQ #"%$ [Note: You may get some weird trigonometric thing (e.g. (+*,ocpq +rs$ ) that you cannot evaluate without your calcular. Just leave it in its trigonometric form (e.g. (+*,Wocpt rsw$ ) in your answer.]

2 p ] A r I ] 2. This problem was mostly done in class. Consider the following system process continuous-time signals with discrete-time processing. Sampler: Freq = Your discrete-time processing olkit on your computer has the following four blocks for possible use (when a frequency response is given, it is valid for F >?@0BA%0DC, and it repeats outside of that range, of course): Block 1: /FG$\& o where you can choose any g that you wish Block 2: /FG$ &F Block 3: /FG$H& Block 4: The fourth block provides output Pc> gqcq&u? ;q$ ;1A K FLK ^ I 3 A else B> gqc for input B> gqc. Using any combinations of these blocks (including multiple versions of any, if needed) and specifying the sampling frequencies 45 and 498, provide a separate system implement each of the following desired continuous-time operations with the block diagram above. /4$ &ST!/4V1.1333$, be applied on input signals! #"%$ of bandwidth up ;1ij2 KHz yield the signal PQ #"%$. (a) The lowpass filter (b) A system that delays the input! #"%$, of bandwidth less than ;=3 KHz, by o r seconds yield the signal Pc #"d$. (c) A system that yields PQ #"%$\&! #"%$ for inputs! #"%$ of bandwidth less than ;=3 KHz. (d) A system that implements the bandpass filter: /4$\& ;1A 21333ZM_K 4HKNM ;= A else on inputs signals! #"%$ of bandwidth less than ;=3 KHz yield the signal PQ #"%$.

3 3. This problem was done in class. In this problem, you are going consider how do multiplication of two continuous-time signals using discrete-time processing. (a) Find the Fourier transform RS /4$ of: and the Fourier transform ab /4$ of! #"%$\& sinc ;=33331"%$ Pc #"d$\& sinc 33331"d$ (b) Find the bandwidth (you do not have find the full Fourier transform unless you would like) of the signal #"%$H& B #"d$ Pc #"%$. (c) Your buddy samples the signal B #"d$ at 15 KHz obtain the sequence B> gqc and the signal Pc #"d$ at 45 KHz obtain the signal Pc> gqc. Give the discrete-time processing of!> gqc and PQ> gqc, and the sampling frequency 418, for the D/C conversion such that the circuit below outputs #"d$x&<! #"%$ Pc #"d$. [Notes: (1) Be sure that your circuit would work for any B #"d$ and PQ #"%$ of the same bandwidth as the signals given. (2) You have all discrete-time blocks for your use, including discrete-time filters of any type, upsamplers, downsamplers, mathematical operations of any type, etc.] Discrete Processing O

4 4. Consider the following system process continuous-time signals with discrete-time processing. / Sampler: Freq = % / Note that there is no anti-aliasing filter on the front-end of the sampler. As in class, let the output of the five blocks be c5q #"%$, B> gqc, PQ> gqc, PW5t #"d$, and PQ #"%$, respectively, with corresponding Fourier transforms given by R 5- /4$, R /FG$, a /FG$, aq5q /4$, and ab /4$. Suppose the input the system is given by:! #"%$\& sinc / "%$ Below you will be asked find a /4$, the Fourier transform of the output PQ #"%$, for a bunch of different systems. Notes: A sketch of ab /4$ is sufficient - no need give an equation. Be sure justify your answers. It is sufficient (and I would encourage you) sketch R 5 /4$, R /F'$, a /FG$, aq5q /4$, and ab /4$, although you can omit Rb5- /4$ and aq5q /4$ if you know what you are doing. Do not worry about amplitudes! (a) Suppose 4 5 & 4 8 & 25 khz, and the discrete-time processing is the filter /FG$H& ;J? K FLK 0 A? 0 MOF O0 (and the filter is periodic with period.10, of course). Find ab /4$, the Fourier transform of the output Pc #"%$. (b) Suppose 45X& 498b& 15 khz, and there is no discrete-time processing (i.e. PQ> gqcg&<!> gqc ). Find a /4$, the Fourier transform of the output Pc #"d$. (c) Suppose you want Pc #"%$\& I #"%$ (output is the square of the input) and you proceed as follows: You let 45e&f.2 khz. Then, realizing you need move a higher sampling rate if you want square, you use the discrete-time processing: (i) interpolate by 8, (ii) decimate by 5, and (iii) Pc> gqc\&) I > gqc. Because of your new sampling rate, you let 48X& 40 khz. Find ab /4$, the Fourier transform of the output PQ #"%$. Also, comment on how well your squarer worked. (d) Suppose you want Pc #"d$h&h I #"d$ (output is the square of the input) and you proceed as follows: You let 4 5 &f.2 khz. Then, realizing you need move a higher sampling rate if you want square, you use the discrete-time processing: (i) decimate by 5, (ii) interpolate by 8, and (iii) Pc> gqc\&)\i1> gqc. Because of your new sampling rate, you let 48X& 40 khz. Find ab /4$, the Fourier transform of the output PQ #"%$.Also, comment on how well your squarer worked.

5 p 5. Consider the following system process continuous-time signals with discrete-time processing. / Sampler: Freq = % / Suppose that the impulse response of our softward module FILTR is given by: > gqcq& 3 ij2 W> g mh;lc m N> gqc`mn3 ij2 N> g? ;lc where W> gqc is the standard discrete impulse function. Note that there is no anti-aliasing filter on the front-end of the sampler. (a) To test our code, we first give input!> gqc\& W> gqc m W> g the output Pc> gqc. Find the output Pc> gqc. (b) Find the frequency response /FG$ of FILTR and plot its magnitude K? -C the module FILTR and obtain /FG$=K (c) Suppose that the continuous-time signal B #"d$ &_(+*,- /.10\;=33331"%$ is input our system, with 4 5J& 498X& 30 khz, and the discrete-time processing is the application of the filter > gqc!> gqc. Find the output PQ #"%$. > & (d) Suppose you could not figure out part (b), so you turn MatLab and the discrete Fourier transform (DFT) help you understand what gqc will do various input frequencies. You are planning on employing 45e&U khz, and you want be able have an analog frequency resolution of at most 40 Hz. What should you employ for your DFT? (e) Your operation from (d) will result in a DFT > ẄC A Z& 3 At;1A=i=i=i-A?b;. At what (approximately) will you find the gain that impacts the sinusoid in part (c)? 6. Consider the following system process continuous-time signals with discrete-time processing. / Sampler: Freq = % / Note that there is no anti-aliasing filter on the front-end of the sampler. Suppose that the input is (+*,-/ "%$, and I want double its frequency, so that the output is (+*,-/ "%$, where is any positive constant. Show how this can be done by the proper p selection of 45 and 498 without any discrete-time processing. discrete-time processing.

6 7. When you sample! #"%$ get the sampled version 5- #"d$, you nearly always get a Fourier transform R 5- /4$ that is non-zero at places where the RS /4$ was zero. This implies that the sampler is not a linear time-invariant (LTI) system. Is it non-linear? Or is it time-variant? (or both). (Be sure start with the definitions of linearity and time-invariance get both get this problem correct and get full credit.)

Multirate Digital Signal Processing

Multirate Digital Signal Processing Basic Sampling Rate Alteration Devices Up-sampler - Used to increase the sampling rate by an integer factor Down-sampler - Used to increase the sampling rate by an integer