Module Code: 7CCSMTDS
Module Name: Topics on Data and Signal Analysis
Module Leader: Zoran Cvetkovic´
Date of Exam: TBC
Exam Weight: 60%
Proportion of exam: Additional Questions (100%)
This exam includes the following question types:
Multiple Choice [yes]: with justification
Other [yes]: uploaded answer
Navigation:
Random [no]
Forward only [no]
The exam consists of multiple choice questions that require justification. To obtain a full mark,
you must select correct answers from provided multiple choices, and upload a written justification
for your answers.
Total length of exam, not including time for uploading working (minutes): 120 minutes
Length of time for uploading working (minutes): 30 minutes
7CCSMTDS
Question One (10 marks): Consider the discrete-time waveforms
ϕ1[n] =

1, n = 0
1, n = 1
0, otherwise
, ϕ2[n] =

1, n = 0
−1, n = 1
0, otherwise
, ϕ3[n] =

1, n = 0
−1
2
, n = 1
0, otherwise
,
and let
Φ1 = {ϕ1[n− k], k ∈ ZZ} , Φ2 = {ϕ2[n− k], k ∈ ZZ} , Φ3 = {ϕ3[n− k], k ∈ ZZ} .
Which of the families Φ1, Φ2, Φ3, is a Riesz basis for `2(ZZ)?
i) Φ1
ii) Φ2
iii) Φ3
iv) all of them
v) none of them
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7CCSMTDS
Question Two (10 marks): Consider a discrete-time waveform
ϕ[n] =

a0, n = 0
a1, n = 1
a2, n = 2
0, otherwise
,
where a0 6= 0, a1 6= 0, a2 6= 0 are real numbers. Assume that a0, a1, a2 are such that
Φ = {ϕ[n− k], k ∈ ZZ}
is a Riesz basis for `2(ZZ) and let Ψ be its dual biorthogonal basis. Are the waveforms in Ψ
of finite or infinite length?
i) they are all of finite length
ii) they are all of infinite length
iii) some are of finite and some of infinite length
iv) their lengths depends on particular values of a0, a1, a2
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7CCSMTDS
H0(z)
H1(z)
H0(z-1)
H1(z-1)
+
Y4,0(z)
X(z) X1(z)
Y3,0(z)Y2,0(z)Y1,0(z)
Y3,1(z)Y2,1(z)Y1,1(z) 22
22
Y4,1(z)
X(!!")
"−$ $−$2 $2
1Figure for Questions Three and Four.Question Three (10 marks): Let the filters H0(z) and H1(z) of the analysis/synthesis filter bank
structure shown in the figure be given in the Fourier domain as
H0(e
jω) =

√
2, |ω| ≤ 1
2
pi
0, 1
2
pi < |ω| < pi
, H1(e
jω) =

0, |ω| ≤ 1
2
pi
−e−jω√2, 1
2
pi < |ω| < pi
If the x[n] is given in the Fourier domain as
X(ejω) =

0, |ω| < 1
4
pi
1 1
4
pi ≤ |ω| ≤ 3
4
pi
0 3
4
pi ≤ |ω| < pi
what is |Y2,0(ejω)| equal to?
i)
|Y2,0(ejω)| =

√
2
2
, |ω| < 1
2
pi
0, 1
2
pi ≤ |ω| < pi
ii)
|Y2,0(ejω)| =

0, |ω| < 1
2
pi
√
2
2
, 1
2
pi ≤ |ω| < pi
iii)
|Y2,0(ejω)| =
√
2
2
, ∀ω
iv)
|Y2,0(ejω)| =

0, |ω| < 1
4
pi
√
2
2
1
4
pi ≤ |ω| ≤ 3
4
pi
0 3
4
pi ≤ |ω| < pi
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7CCSMTDS
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7CCSMTDS
H0(z)
H1(z)
H0(z-1)
H1(z-1)
+
Y4,0(z)
X(z) X1(z)
Y3,0(z)Y2,0(z)Y1,0(z)
Y3,1(z)Y2,1(z)Y1,1(z) 22
22
Y4,1(z)
X(!!")
"−$ $−$2 $2
1Figure for Questions Three and Four.Question Four (10 marks): Let the filters H0(z) and H1(z) of the analysis/synthesis filter bank
structure shown in the figure be given in the Fourier domain as
H0(e
jω) =

√
2, |ω| ≤ 1
2
pi
0, 1
2
pi < |ω| < pi
, H1(e
jω) =

0, |ω| ≤ 1
2
pi
−e−jω√2, 1
2
pi < |ω| < pi
If the x[n] is given in the Fourier domain as
X(ejω) =

0, |ω| < 1
4
pi
1 1
4
pi ≤ |ω| ≤ 3
4
pi
0 3
4
pi ≤ |ω| < pi
what is |Y3,1(ejω)| equal to?
i)
|Y3,1(ejω)| =

√
2
2
, |ω| < 1
2
pi
0, 1
2
pi ≤ |ω| < pi
ii)
|Y3,1(ejω)| =

0, |ω| < 1
2
pi
√
2
2
, 1
2
pi ≤ |ω| < pi
iii)
|Y3,1(ejω)| =
√
2
2
, ∀ω
iv)
|Y3,1(ejω)| =

0, |ω| < 1
4
pi
√
2
2
1
4
pi ≤ |ω| ≤ 3
4
pi
0 3
4
pi ≤ |ω| < pi
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7CCSMTDS
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7CCSMTDS
Ha(z)
Hb(z)
A
B
Hc(z)
Hd(z)
C
D
x[n]
ya[n]
yb[n]
yc[n]
yd[n]
H1(z)
H0(z)
2
2
H1(z)
H0(z)
2
2
H1(z)
H0(z)
2
2
x[n]
yb[n]
yc[n]
yd[n]
ya[n]
Figure for Questions Five, Six and Seven.
Question Five (5 marks): Consider the filter bank tree shown in the figure (top) and its equiva-
lent four channel filter bank as shown in the figure (bottom). What are the decimation ratios
A, B, C and D of the equivalent filter bank?
i) A = 2, B = 8, C = 8, D = 4
ii) A = 2, B = 4, C = 8, D = 8
iii) A = 2, B = 4, C = 6, D = 8
iv) A = 2, B = 4, C = 4, D = 8
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7CCSMTDS
Ha(z)
Hb(z)
A
B
Hc(z)
Hd(z)
C
D
x[n]
ya[n]
yb[n]
yc[n]
yd[n]
H1(z)
H0(z)
2
2
H1(z)
H0(z)
2
2
H1(z)
H0(z)
2
2
x[n]
yb[n]
yc[n]
yd[n]
ya[n]
Figure for Questions Five, Six and Seven.
Question Six (10 marks): Consider the filter bank tree shown in the figure (top) and its equiva-
lent four channel filter bank as shown in the figure (bottom). Let H0(z) be an ideal half-band
low-pass filter and H1(z) be an ideal half-band high-pass filter. What kind of filter Hb(z) is?
i) an ideal band-pass filter with the pass band (1
8
pi, 3
8
pi)
ii) an ideal band-pass filter with the pass band (1
4
pi, 1
2
pi)
iii) an ideal band-pass filter with the pass band (1
8
pi, 1
4
pi)
iv) an ideal band-pass filter with the pass band (1
4
pi, 3
8
pi)
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7CCSMTDS
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7CCSMTDS
Ha(z)
Hb(z)
A
B
Hc(z)
Hd(z)
C
D
x[n]
ya[n]
yb[n]
yc[n]
yd[n]
H1(z)
H0(z)
2
2
H1(z)
H0(z)
2
2
H1(z)
H0(z)
2
2
x[n]
yb[n]
yc[n]
yd[n]
ya[n]
Figure for Questions Five, Six and Seven.
Question Seven (10 marks): Consider the filter bank tree shown in the figure (top) and its
equivalent four channel filter bank as shown on the figure (bottom). Let
H0(z) =
1√
2
(1 + z−1) , H1(z) =
1√
2
(1− z−1)
What is Hc(z) equal to?
i) Hc(z) =
√
2
4
(1 + z−1 + z−2 + z−3 + z−4 + z−5 + z−6 + z−7)
ii) Hc(z) =
√
2
4
(1 + z−1 + z−2 + z−3 − z−4 − z−5 − z−6 − z−7)
iii) Hc(z) =
√
2
4
(1 + z−1 − z−2 − z−3 + z−4 + z−5 − z−6 − z−7)
iv) Hc(z) =
√
2
4
(1 + 2z−1 + 3z−2 + 4z−3 − 4z−4 − 3z−5 − 2z−6 − z−7)
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7CCSMTDS
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7CCSMTDS
Question Eight (10 marks): Consider the wavelet series expansion of continuous-time signals
and assume that ψ(t) is the Haar wavelet,
ψ(t) =

1, 0 ≤ t < 1
2−1, 1
2
≤ t < 1
0, otherwise
.
Recall that {ψmn(t) : ψmn(t) = 2−m2 ψ(2−mt − n), m ∈ ZZ, n ∈ ZZ} is an orthonormal basis
for L2(IR). What are the expansion coefficients of
f(t) = ψ
(
t
2
+
1
2
)
i)
〈ψmn(t), f(t)〉 =

−2−m2 , m > 0, n = −1
−2−m2 , m > 0, n = 0
0, otherwise
ii)
〈ψmn(t), f(t)〉 =

−2−m2 , m ≤ 0, n = −1
−2−m2 , m ≤ 0, n = 0
0, otherwise
iii)
〈ψmn(t), f(t)〉 =

−2−m+22 , m > −1, n = −1
−2−m+22 , m > −1, n = 0
0, otherwise
iv)
〈ψmn(t), f(t)〉 =
{ √
2, m = 1, n = −1
0, otherwise
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7CCSMTDS
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7CCSMTDS
Question Nine (10 marks): Consider the wavelet series expansion of continuous-time signals
and assume that ψ(t) is the Haar wavelet,
ψ(t) =

1, 0 ≤ t < 1
2−1, 1
2
≤ t < 1
0, otherwise
Recall that {ψmn(t) : ψmn(t) = 2−m2 ψ(2−mt − n), m ∈ ZZ, n ∈ ZZ} is an orthonormal basis
for L2(IR). Consider signal f(t) = ϕ(t)− ϕ(t− 1) where
ϕ(t) =
{
1, 0 ≤ t < 1
0, otherwise.
What is ∑
m∈ZZ
∑
n∈ZZ
|〈ψmn(t), f(t)〉|2
equal to for this signal?
i) 1
2
ii) 1
iii)
√
2
iv) 2
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7CCSMTDS1(t)
−1 1−12 12
12
−12
2(t)
− 12 12
12
Figure for Question 10.
Question 10 (10 marks): Consider selecting a wavelet function for the continuous wavelet
transform. Which of the functions shown in the figure is an admissible wavelet for the
continuous wavelet transform.
i) neither ψ1(t) nor ψ2(t)
ii) ψ1(t)
iii) ψ2(t)
iv) both ψ1(t) and ψ2(t)
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7CCSMTDS!(t)
%−1 1−12 12
12
−12
Figure for Question 11.
Question Eleven (5 marks): Consider the continuous wavelet transform (CWT) of a signal f(t)
using wavelet ψ(t) shown in the figure. Recall that the CWT of a signal f(t) ∈ L2(IR) is
defined as
CWTf (a, b) =
∫ ∞
−∞
ψ∗(a, b)f(t)dt = 〈ψa,b(t), f(t)〉 , a ∈ IR, b ∈ IR (1)
where
ψa,b(t) =
1√
|a|
ψ
(
t− b
a
)
. (2)
What is the region in the a, b plane where CWTf (a, b) is influenced by the value of f(t) at
some time instant t = t0.
i) t0 − 12a < b < t0 + 12a
ii) 1
2
(t0 − a) < b < 12(t0 + a)
iii) t0 − a < b < t0 + a
iv) a− 1
2
t0 < b < a+
1
2
t0
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7CCSMTDS
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17