ECE551 Advanced Topics in Digital Signal Processing Homework #1 Due September 17, 2009 Review undergraduate material on discrete–time signals and systems, and the DFT. Recom- mended reading: Proakis and Manolakis, chapters 1–6; or Oppenheim and Shafer, chapters 1–5, 8, 9, 11.1 and 11.2. Problem 1. Compute the L p norm of x(t)= e −t/2 :0 ≤ t< 10 0 : else. Evaluate the resulting expression for p = 1, p = 2 and p →∞. Problem 2. (a) Evaluate the expressions δ(2t) x(t) dt, δ(t 1/3 ) x(t) dt, and δ(t 2 − 1) x(t) dt. (b) Derive an expression for δ(f (t)) x(t) dt, where the function f (t) is arbitrary but contin- uous everywhere on the real line. Problem 3. A square pulse x(t) with amplitude and width equal to 1 is filtered through a LTI system with frequency response H (F ). Sketch the output ˆ x(t) for each of the three systems below, assuming that B = 10 Hz. Your sketches should be as realistic as possible and should clearly show (1) possible smoothing of the edges of the pulse, (2) overshoots, and (3) ripples in ˆ x(t). Compare the merits of the three systems with regard to criteria (1), (2) and (3). -B B F(Hz) 1 0 H(F) H(F) F(Hz) 0 1 -B B H(F) F(Hz) 0 B -B 1 sin ( PI * F / B) PI * F / B 1
Transcript
ECE551
Advanced Topics in Digital Signal Processing
Homework #1
Due September 17, 2009
Review undergraduate material on discrete–time signals and systems, and the DFT. Recom-mended reading: Proakis and Manolakis, chapters 1–6; or Oppenheim and Shafer, chapters1–5, 8, 9, 11.1 and 11.2.
Problem 1. Compute the Lp norm of x(t) ={
e−t/2 : 0 ≤ t < 100 : else.
Evaluate the resulting expression for p = 1, p = 2 and p → ∞.
Problem 2.
(a) Evaluate the expressions∫
δ(2t)x(t) dt,∫
δ(t1/3)x(t) dt, and∫
δ(t2 − 1)x(t) dt.
(b) Derive an expression for∫
δ(f(t))x(t) dt, where the function f(t) is arbitrary but contin-uous everywhere on the real line.
Problem 3. A square pulse x(t) with amplitude and width equal to 1 is filtered through a LTIsystem with frequency response H(F ). Sketch the output x(t) for each of the three systemsbelow, assuming that B = 10 Hz. Your sketches should be as realistic as possible and shouldclearly show (1) possible smoothing of the edges of the pulse, (2) overshoots, and (3) ripplesin x(t). Compare the merits of the three systems with regard to criteria (1), (2) and (3).
-BB
F(H
z)
1 0H(F
) H(F
)
F(H
z)
01
-BB
H(F
)
F(H
z)
0
B-B
1
sin
( PI
* F
/ B
)
PI *
F /
B
1
Problem 4. The output y(t) of a linear time-invariant system with input x(t) = e−tu(t) isgiven by y(t) = e−t2u(t). Determine the impulse response of that system. Is the system BIBOstable?
Problem 5. Compute the lp norm of the sequence x(n) ={
e−n/2 : 0 ≤ t < 100 : else.
Evaluate the resulting expression for p = 1, p = 2 and p → ∞.
Problem 6. How many continuous derivatives does the DTFT of x(n) = 11+|n|4 have?
Problem 7. Determine all possible stable sequences that have the following z–transform:X(z) = z−1
(1−3z−1)(1−2z−1)2. Indicate the location of the zeros and poles of X(z). Which one(s)
of these sequences have a DTFT?
Problem 8. Determine the ROC of X(z) =∑∞
n=−∞ sinc(n/2) z−n.
Problem 9. The Fourier transform of some real, continuous-time signal xa(t) is zero atfrequencies greater than B. The classical sampling theorem asserts that xa(t) may be recon-structed from its samples taken at the rate 1
2B . Here we want to show that xa(t) may also bereconstructed from its samples taken the rate T = 1
B , so long as both the amplitude and theslope of xa(t) are measured at each sampling instant. Show that the reconstruction formula is
xa(t) =∞∑
n=−∞xa(nT ) a(t − nT ) +
∞∑n=−∞
x′a(nT ) b(t − nT ),
using the two interpolating functions a(t) = sinc2(t/T ) and b(t) = t sinc2(t/T ).
Problem 10. Under what conditions can a signal xa(t) be reconstructed from measurementsof its amplitude and all of its derivatives at time t = 0?
Problem 11. In class we have discussed conditions for existence of the Fourier transform. Wehave also mentioned that it would be nice to extend the definition of the transform to signalssuch as pure sine waves, for which the Fourier integral does not converge. In this problem,you will show how to do this. Consider the constant signal x(t) ≡ 1 defined for −∞ < t < ∞.We are tempted to write its Fourier transform as X(f) =
∫ ∞−∞ e−j2πft dt. Unfortunately, such
integrals cannot be interpreted in the usual sense, see footnote in Lecture 2 class notes, p. 5.Derive an expression for X(f) using the following steps:
1. Define a sequence of “well-behaved” signals xσ(t) such that limσ→∞ xσ(t) = x(t).
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGNDepartment of Electrical and Computer Engineering
ECE 551 Digital Signal Processing II
Spring 2006
Problem Set 2
Issued: Thursday, January 26, 2005 Due: Thursday, February 2, 2005
Reading: Lecture notes 1-4, and Chapter 5 in DSP (Oppenheim et al.).
Problem 2.1
Consider the causal discrete-time system with impulse response h[n] and system function
H(z) = e1/z
(a) Determine the unit pulse response h[n] of the system.
(b) Determine whether or not H(z) corresponds to a minimum phase system.
Problem 2.2
One of the interesting and important properties of minimum-phase sequences is the minimum-energy delay property, i.e., of all the causal sequences having the same Fourier transform magnitudefunction, |H(ejω)|, the quantity
E[n] =
n∑m=0
|h[n]|2
is maximum for all n ≥ 0 when h[n] is the minimum-phase sequence. This result is proved asfollows: Let hmp[n] be a minimum-phase sequence with z-transform Hmp(z). Furthermore, let zk
be a zero of Hmp(z) so that we can express Hmp(z) as
Hmp(z) = Q(z)(1 − zkz−1), |zk| < 1,
where Q(z) is again minimum-phase. Now consider another sequence h[n] with z-transform H(z)such that
|H(ejω)| = |Hmp(ejω)|
and such that H(z) has a zero at z = 1/z∗k instead of at zk.
(a) Express H(z) in terms of Q(z).
(b) Express h[n] and hmp[n] in terms of the minimum-phase sequence q[n] that has z-transformQ(z).
Problem Set 2 2
(c) To compare the distribution of energy of the two sequences, show that
ǫ[n] =
n∑m=0
|hmp[m]|2 −
n∑m=0
|h[m]|2 = (1 − |zk|2)|q[n]|2.
(d) Using the result of part (c), argue that
n∑m=0
|h[m]|2 ≤n∑
m=0
|hmp[m]|2, for all n.
Problem 2.3
Consider the discrete-time LSI system with frequency response magnitude satisfying, |H(ejω)|2 =54
+ cos(ω), for |ω| < π.
(a) Find the impulse response of the minimum phase system hmp[n] corresponding to this fre-quency response magnitude, i.e. |Hmp(e
jω)|2 = 54
+ cos(ω), |ω| < π.
(b) Let B(z) = (12
+ z−1). Show that |B(ejω)|2 = 54
+ cos(ω) as well.
(c) Determine the real-valued all-pass system G(z) such that B(z) = G(z)Hmp(z), where Hmp(z)is the system function for the minimum-phase system from part (a).
(d) Determine∑
∞
n=−∞g[n]g[n + m].
Problem 2.4
Consider a discrete-time system with frequency response magnitude |H(ejω)|2 which satisfies|H(ejω)|2 = G(ejω) where G(ejω) is the discrete-time Fourier transform of the sequence g[n] and
g[n] = −1
2δ[n − 1] +
5
4δ[n] −
1
2δ[n + 1]
(a) Express g[n] in terms of h[n]. Provide as simple an expression as you can.
(b) Can h[n] be determined directly from g[n]? If so, find h[n]. If not, provide a mathematicaldescription of all possible sequences h[n] consistent with this information.
(c) If it is known that h[n] is a minimum-phase sequence, can h[n] be determined directly fromg[n]? If so, find h[n]. If not, provide a mathematical description of all possible sequences h[n]consistent with this information.
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGNDepartment of Electrical and Computer Engineering
ECE 551 Digital Signal Processing II
Spring 2006
Problem Set 3
Issued: Thursday, February 2, 2005 Due: Thursday, February 9, 2005
Problem 3.1
Prove the formulas (discussed in class) for conversion between FIR lattice structure and directform.
Problem 3.2
The direct form flow graph for a causal, LTI system is shown below:
x[n]
y[n]
z -1 z -1 z -1
-1.5 -1 0.5
(a) Draw the lattice form flow graph for an equivalent system, i.e. for a system which has thesame impulse response.
(b) Verify that the two systems have the same impulse response by tracing an impulse inputthrough all of the paths in the flow graphs and summing the impulses that arrive at theoutput at the same delay.
Problem 3.3
The lattice network shown in (A) below has two inputs x1[n]) and x2[n] and two outputs y1[n]and y2[n]. If the output y1[n] in (A) is connected to the input x2[n], we obtain the network in (B).
Problem Set 3 2
z−1
x [n] 1
x [n] 2
[n]y2
1[n]y
−K
K
(A)
z−1
x [n] 1
x [n] 2
[n]y2
1[n]y
−K
K
(B)
(a) Write the two difference equations relating the two inputs to the two outputs for the networkin (A).
(b) Determine the system transfer function H11(z) = Y1(z)/X1(z) for the network in (B). Forwhat values of k will the system be stable?
(c) Determine the system transfer function H21(z) = Y2(z)/X1(z) for the network in (B). Plot|H21(e
jω)| for |ω| < π.
Problem 3.4
Show that an even-length, symmetric linear-phase FIR filter cannot be a true high-pass filter;that is, show that H(ejπ) = 0 for all choices of coefficients. For all four types of linear-phase FIRfilters (even and odd length, symmetric and anti-symmetric), determine whether or not each typecan be a true lowpass or highpass filter.
Problem 3.5
Approximation accuracy for FIR filter design is often measured by the energy of the approx-imation error H(ω) − Hd(ω). Show that the window that minimizes this energy criterion is therectangular window.
Problem 3.6
Differentiator filter design:
(a) Design a length-7 differentiator filter, which has desired generalized amplitude response
Hd(ω) = jω , − π < ω ≤ π
using the window design method using a Hamming window. (Be sure to incorporate linearphase in your design.) Is the resulting design desirable? Do you expect substantially differentresults using frequency sampling design?
(b) Design a length-8 differentiator filter using the frequency sampling design method with equallyspaced samples ωk = 2πk/8, using whichever symmetry seems most appropriate.
tQu�w[ymza|O}à�g C5D2J��>�AG(*-�+82B+89@9:)[+8�����=�O�=/=(30Ø��>:/��������=/=(30á7aF,2J;</�>:5D2�§�hÓd�q Ó £ �Hâ sOãäMxK
N â sOãäMxKDÓ £ � ã�æåJ5���I8-?+.),�j7c58-�+82$>@0E),9@(30E(324/�+./�>@5D2Ã587�/��G(È�=����/=(30±>@�6���J5o��2�>@2��{>:IDFJ-�(8°�N.°�h+4��'�(*/=(*-?01>@2J(Q/��G(_;<5d(*ç�;*>@(324/��½è�pUéxp%+82JABê,p¸��FJ;?�B/��J+./�/��G(QåJ5��ëI8-º+.),�$>�2B�{>:IG°QN.°�>���+Í-�(3+89@>:ì3+./�>:5D2B587�§�hÓd�º°�a�U��� 2½+�)J-?+8;</�>@;*+894>@0E),9:(301(324/�+./�>:5D2�587�/��G(qå,5o�íI8-?+.)[�¸po/��G(;<5d(*ç�;*>@(324/��*p8è�pxéxpx+82JAvêJp.01>:ID�d/¸�%(q�dF,+824/�>:ì*(3A���½-�5DFJ2JAJ>�2GI�/��G(C(<îO+8;</m�x+89�FG(3�X/=5�/��G(�2G(3+.-�(3�=/m/=(32d/��¸p8>�°Ñ(8°msOãäMxK��C5DFJ9@AQ�U(-�5DFJ2JAJ(3Av/=5�sOãäM�+82JA�N.ãðï.ñ8»ON�C5DFJ9@Av�%(q-?5DFJ2JAG(3AQ/=5_N.ãäò�°{ó¡5DFJ9@AQ/��G(C-�(3��FJ9:/�>�2GI6�=�O�=/=(30ë��/�>@9@98�%(+82_+89@9:)[+8���X�=�O�=/=(30æ+.7c/=(*-{-�5DF,2JAJ>@2GIdô
ú ³ ´%µ sOãäMxK,� ú ³ ´ â N µ â¡û ³ ´%µ � Ì û ³ ´ â N µ ã�h;o��'�-?+3��/��J(åJ5��§I8-?+.),�_587[+�2G(*/��C58-�Ä_/��J+./m-�(3�dF,>:-�(3��/��C5vAG(39@+3�Q(39:(30E(32d/��{�,FG/X5D2,9:�v5D2G(�0ÈF,9:/�>:),9@>�;*+./�>:5D2����+1;<5D2,�=/�+824/�58/��G(*-�/��J+82�üSN.°�hA[��óý>:/��j��FJ+824/�>@ì*(3Aj;<5d(*çÍ;*>:(324/��3pJ�5DF,9@AÃ/��G(Q2G(*/þ�58-?Ä�587{),+.-?/Q�h;o�C�%(v+82j+89@9:),+8�?���=����/=(30Íô
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGNDepartment of Electrical and Computer Engineering
ECE 551 Digital Signal Processing II
Spring 2006
Problem Set 4
Issued: Thursday, February 9, 2005 Due: Thursday, February 16, 2005
Problem 4.1
Design a length-2 least-squares optimal anti-symmetric FIR filter for a highpass filter withωs = π/3, ωp = π/2. The weight function is 0 in the transition band, 1 in the passband, and 2 inthe stopband. Sketch the frequency response of your filter.
Problem 4.2
In this problem we shall design a high-pass FIR filter by minimising the weighted L2 error, i.e.,
h∗[n] = argminh[n] ǫ2 =1
2π
∫ π
−πW (ω)|Hd(e
jω) − H(ejω)|2dω
The specifications are given below:
(1) M = 10
(2) Hd(ejω) =
{e−jωM/2, π
2< |ω| ≤ π
0, else
(3) W (ω) =
10, |ω| < π3
1, π2
< |ω| ≤ π0, else
(a) Solve the integrals (refer Lecture 16 notes) to get d[n] and Mm,n.
(b) Use Matlab backslash operator to find h∗[n].
(c) Plot H(ejω).
(d) Verify using the command firls.
Problem Set 4 2
Problem 4.3
In this problem, you will design a length three FIR filter to approximate the following frequencyresponse:
Hd(ejω) = jω, |ω| < π
by minimizing the following error criterion,
E =1
2π
∫ π
−πW (ω)|Hd(e
jω) − H(ejω)|2dω,
for
W (ω) =1
| sin(ω)|, 0 < |ω| ≤
π
2
(a) Let h[n] = aδ[n + 1] + bδ[n] + cδ[n − 1] be your length three filter, where a, b, and c arereal constants. Show that in order to minimize E above, that b = 0 and c = −a. Note thatH(ejω) is the discrete-time Fourier transform of h[n].
(b) Find the filter of the form h[n] = aδ[n + 1] − aδ[n − 1] that minimizes E.
Problem 4.4
Design a length-3, symmetric equiripple high-pass filter with the following Ad(ejω) with a stop-
band edge of π3
and a passband edge of π2, and with uniform weighting of the stopband and passband.
What are the filter coefficients and the ripple amplitude δ?
Ad(ejω) = 1,
π
2≤ ω ≤ π (1)
= 0, otherwise. (2)
Hint: You do not need Parks-McCellan algorithm to solve this problem.
Problem 4.5
Problem 7.36a-f in Oppenheim and Schafer (2nd Ed.)
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGNDepartment of Electrical and Computer Engineering
ECE 551 Digital Signal Processing II
Spring 2006
Problem Set 5
Issued: Tuesday, February 23, 2006 Due: Thursday, March 2, 2006
Reading: Lecture notes 13-18, Chp. 7 (Oppenheim et al.), and Chp. 8 ( Proakis et al.)
Problem 5.1
In this problem, you will use Prony’s method (sometimes called the Extended Prony’s method)to find a first-order approximation to the IIR filter with impulse response hd[n] and system function
Hd(z) =1
(1 − 12z−1)(1 − 1
4z−1)
.
That is, you will determine the coefficients b0, b1, and a1 in
H(z) =b0 + b1z
−1
1 + a1z−1≈ Hd(z),
in a quasi-L2 method. Note that Hd(z) satisfies the following difference equation:
hd[n] +
N−1∑k=1
akhd[n − k] =
{bn, 0 ≤ n ≤ M − 10, else
,
where N = 3, and M = 2 for this second-order filter. Prony’s method uses this relationship betweenthe impulse response and the filter coefficients to find the approximating filter by setting N andM to the denominator and numerator orders of H(z), the approximating filter, and then solvingfor ak and bk in a least-squares fashion.
(a) First, determine hd[n] and set b0 = hd[0].
(b) The second step is to solve for the coefficients ak which minimize the expression
∞∑n=M
|hd[n] +N−1∑k=1
akhd[n − k]|2.
Verify that the minimizing ak’s satisfy:
N−1∑k=1
ak
∞∑n=M
hd[n − k]hd[n − ℓ] = −
∞∑n=M
hd[n]hd[n − ℓ], for ℓ = 1 . . . N − 1.
Problem Set 5 2
In this case, you have M = 2, and N = 2, so this reduces to the single equation:
a1
∞∑n=2
hd[n − 1]hd[n − 1] = −∞∑
n=2
hd[n]hd[n − 1],
which you must solve for the coefficient a1. Use this equation to solve for a1.
(c) The last step is to solve for the remaining numerator coefficients, bn. Verify that the coeffi-cients bn should satisfy:
bn = hd[n] +
N−1∑k=1
akhd[n − k].
Use this expression to solve for the coefficient b1.
(d) Use the Matlab function prony to verify that you have the correct coefficients. Plot theimpulse response of the filter you obtained and hd[n]. Is it a good match?
(e) Repeat this problem, using Shanks method. Which has a better impulse response match?
Problem 5.2
Given a desired impulse response hd[n], Prony’s method solves for the set of filter coefficients(numerator coefficients bk, k = 0, . . . , (M − 1) and denominator coefficients ak, k = 1, . . . , (N − 1))which match the first M samples of the impulse response and minimize a least-squares criterionover the remaining samples of the impulse response. The criterion which is minimized is
E =
N0∑n=M
|hd[n] +
N−1∑k=1
akhd[n − k]|2,
where N0 can be finite if the approximation is desired to be based on a finite-length segment of theimpulse response. We can define an error sequence e[n] which is minimized,
E =
N0∑n=M
|e[n]|2.
where
e[n] = hd[n] +
N−1∑k=1
akhd[n − k], M ≤ n ≤ N0
= hd[n] +
N−1∑k=1
akhd[n − k] − bn, 0 ≤ n ≤ (M − 1)
Problem Set 5 3
(a) Let h[n] be the impulse response of the filter designed with Prony’s method, i.e. the filterwhose system function is given by
H(z) =
M−1∑k=0
bkz−k
1 +
N−1∑k=1
akz−k
.
Determine an expression for h[n], n ≥ 0 in terms of the filter coefficients ak, k = 1, . . . ,N − 1and bl, l = 0, . . . ,M − 1.
(b) Let ǫ[n] = hd[n] − h[n], n ≥ 0 be the impulse response approximation error. Show that e[n]and ǫ[n] satisfy
ǫ[n] = e[n] −
N∑k=1
akǫ[n − k]
Problem 5.3
Consider the discrete-time LTI system with input x[n] and output y[n] satisfying
(a) Determine the system function of the whitening filter and its poles and zeros.
(b) Determine the power density spectrum of {x(n)}.
Problem 6.5
An AR(2) process ie defined by the following difference equation
x[n] = x[n − 1] − 0.6x[n − 2] + w[n]
where w[n] is a white noise process with variance σ2w. Use the Yule-Walker equation to solve for
the values of the autocorrelation γxx(0), γxx(1), and γxx(2).
Problem 6.6
Use the orthogonality principle to determine the normal equations and the resulting minimumMSE for a forward predictor of order N that predicts m samples (m > 1) into the future (m-stepforward predictor). Sketch the prediction error filter.
