GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING

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1 GEORGIA INSIUE OF ECHNOLOGY SCHOOL of ELECRICAL and COMPUER ENGINEERING ECE 6250 Spring 207 Problem Set # his assignment is due at the beginning of class on Wednesday, January 25 Assigned: 6-Jan-7 Due Date: 25-Jan-7 As stated in the syllabus, unauthorized use of previous semester course materials is strictly prohibited in this course PROBLEM : Using you class notes, prepare a -2 paragraph summary of what we talked about in class in the last week I do not want just a bulleted list of topics, I want you to use complete sentences and establish context (Why is what we have learned relevant? How does it connect with other things you have learned here or in other classes?) he more insight you give, the better PROBLEM 2: Consider the analog system below x(t) H c (j ) y c (t) where H c (jω) = 000π + jω We wish to implement a (near) equivalent system in digital using the following architecture: x(t) H a (j ) C/D H d (e j! ) y[n] D/C y d (t) where = /3000 and H a (jω) is an antialiasing filter; H a (jω) =, Ω 3000π and is 0 otherwise Assume the D/C block incorporates an ideal reconstruction filter Find H d (e jω ) such that y d (t) matches y c (t) as closely as possible Sketch your answer

2 X c (jω) 5 5π 5π Ω x c (t) C/D D/C x r (t) =/4 =/4 PROBLEM 3: Suppose that a continuous-time signal x c (t) has the Fourier transform shown below: he signal is used as the input to two different systems, one without an anti-aliasing filter: and one with an anti-aliasing filter: Above, C/D is an ideal continuous-to-discrete converter, D/C is an ideal discrete-to-continuous converter, and Ω 4π H a (jω) = 0 else Calculate the energy of the reconstruction errors in both cases: x c (t) x r (t) 2 2 and x c (t) x r (t) 2 2, where f(t) 2 2 = f(t) 2 dt 2 Argue (rigorously) that there is no other signal that could come out of the D/C converter that is closer to x c (t) than x r (t) (By closer, we mean that the energy in the error is smaller) Knowledge of the Parseval heorem will definitely help for both parts (a) and (b): for continuous time signnal x(t) with CF X(jΩ), x(t) 2 dt = 2π X(jΩ) 2 dω 2

3 x c (t) H a (jω) C/D D/C x r (t) =/4 =/4 PROBLEM 4: Let x c (t) be a continuous-time signal which is bandlimited to π/, and let correspond to samples taken at the Nyquist rate: = x c (n ) As we know, x c (t) can be perfectly reconstructed from using sinc interpolation, x c (t) = n= sin(π(t n )/ ) π(t n )/ his reconstruction process can be modeled as converting into a spike train, and then passing it through an ideal lowpass filter Suppose instead we generate a continuous-time signal using a different interpolation kernel: where x r (t) = φ(t) = n= φ(t n ), t t 0 t > We will use FOH to denote the system that maps to x r (t): FOH x r (t) Sketch φ(t) Sketch x r (t) for = δ[n + ] + 3δ[n] δ[n ] + δ[n 2] + 2δ[n 3], where δ[n] is the Kronecker delta function (δ[n] = for n = 0 and is zero for all other values of n) How would you describe the action of FOH? 2 We would like to model FOH as conversion to a spike train followed by an analog filter: What must the frequency response H a (jω) be? 3

4 x H a (j ) x r (t) X n= (t n ) FOH 3 Now we precondition the samples by passing them through a digital filter, and postcondition the output of FOH by passing it through an analog low-pass filter (see the figure below): Ω π/ H LP = 0 otherwise Find H d (e jω ) so that x r (t) = x c (t) H d (e j! ) FOH H LP (j ) x r (t) PROBLEM 5: Let be a discrete-time signal whose DF X(e jω ) = 0 for π/3 ω π which we wish to convert to analog using a standard D/C converter Suppose that right before we feed this signal into the converter, and adversary changes exactly one sample of is an unknown way at an unknown location n 0, forming n n 0 = something else n = n 0 Given our knowledge that the original DF X(e jω ) is zero for π/3 ω π, describe how such an error might be detected by examining the output y(t) 2 Describe (and sketch a diagram of) a method to correct the error Hint: one way to do this is with C/D converters which take equally spaced samples but at different offsets hat is, suppose the C/D system takes samples with spacing but offset τ, so = x c (τ + n ) 4

5 PROBLEM 6: In class last week, we reviewed the fact a sinc in the continuous-time domain transforms into a block in the Fourier domain: h (t) = sin(πt/ ) πt/ CF H (jω) =, Ω π, 0, Ω > π In this problem, we derive the analogous expressions for the other three types of transforms and use MALAB to plot the results Suppose that a discrete-time signal h[n] has discrete-time Fourier transform (DF) H(e jω ) = h[n] e jωn M ω π/m = 0 π/m < ω π n= Find h[n] using the inversion formula h[n] = 2π π π H(e jω ) e jωn dω Use the stem command in MALAB to plot it for M = 5 2 Suppose that a continuous-time signal x(t) on the interval [ /2, /2] has Fourier series /2 α k = x(t)e j2πkt dt = 2B+ k = B,, 0,, B 0 otherwise /2 Find x(t) using the synthesis formula x(t) = k= α k e j2πkt Use the plot command in MALAB to plot it for B = 7 3 Suppose that a vector x R N has discrete Fourier transform (DF) ˆx[k] = N n=0 Find using the synthesis formula e j2πkn/n 2B+ k = 0,, B = 0 k = B +,, N B 2B+ k = N B,, N = N k=0 ˆx[k] e j2πkn/n Use the stem command in MALAB to plot it for N = 0 and B = 5 (Of course the answers to these questions are widely available, but you should actually work it out) 5

6 PROBLEM 7: Suppose that x is a column vector in MALAB with 0 components segment of code: Consider the following fx = fft(x); fxi = 0*[fx(:5); zeros(909,); fx(52:0)]; xi = ifft(fxi); he 00 vector xi is an interpolation of x o see this, you can set x = randn(0,); run the code above, and then plot the results: plot(xi) hold on stem(:0:00,x, r ) Find an explicit expression for the interpolation formula his should take the form of 0 x i [n] = x[k] (something that depends on n and k) k= 6