department of electrical and computer engineering, …dllamocca.org/ece-314/solutions - homework...

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, THE UNIVERSITY OF NEW MEXICO

ECE-314: Signals and Systems Summer 2013

Instructor: Daniel Llamocca

Solutions - Homework # 4

PROBLEM 1 Find the FT representation of the following periodic signals: a) b) c)

d)

e)

a) Periodic signal FS, T = 4,

Thus:

Finally:

x(t)

t

......

1-1

-1

1

3

x(t)

t

......

1

1 3-1




b) Periodic signal FS, T = 2,

Thus:

Finally:




c) . Periodic signal FS

Thus:

Finally:

d) . Periodic signal FS,

Thus:

Finally:

e)

. Period Signals FS.

Finally:

x(t)

t

1

... ...

2 4




PROBLEM 2 Find the time-domain periodic signal corresponding to the following FT representations:

a)

b)

c)

d)

Process: i) Get the FS coefficients , ii) plug them in the equation for .

a)

Then:

Finally:

b)

Then:

Finally:

c)

Then:

Finally:




d)

Then:

Finally:

PROBLEM 3 The DTFT formula for periodic signals is usually specified as:

Since is N-periodic, a simpler way to specify the DTFT is to specify it only over one period ( ):

Once you specify over one -period, all you need is to generate infinite periodic replicas. a) Given the following DTFS coefficients for one period ( ), sketch the DTFT

representation for one -period. Then, generate the DTFT representation for all periods by generating infinite replicas on both sides. N = 4. Notice that since , and , then

4

X[k]

k

1 2 30-3 -2 -1

1

2

3




b) Now, you are given the DTFS coefficients for one period ( ). Sketch the DTFT representation for one -period. Then, generate the DTFT representation for all periods by generating infinite replicas on both sides. Notice that since , and , then (a range)

c) The two DTFT representations of (a) and (b) must be the same. Actually, you could pick any N-period in and get the same DTFT. For example, sketch for one period ( ), and show that the DTFT representation is the same as those of (a) and (b). Notice that since , and , then (a range)

d) Given the following DTFS coefficients for one N-period ( ), sketch for all periods.

Then, redefine one period of for the following periods: and

.

e) When referring to the DTFT over one period, it is customary to pick , or .

For N = 7 and 8, demonstrate that: i. If , and , then .

ii. If

, and , then .

a)

. For one period:

3

X[k]

k

0 1 2-3 -2 -1

1

3

2

4

4

X[k]

k

1 2 30-3 -2 -1

1

2

3

0

X(ej )

2p

4p

6p

p/2 p 3p/2-3p/2 -p -p/2-2p 2p 5p/2 3p 7p/2

......




b)

c)

d) Period: (with the infinite copies to both sides)

0

X(ej )

2p

4p

6p

p/2 p 3p/2-3p/2 -p -p/2-2p 2p 5p/2 3p

......

-5p/2

4

X[k]

k

1 2 30-3 -2 -1

1

2

3

0

X(ej )

2p

4p

6p

p/2 p 3p/2-3p/2 -p -p/2-2p 2p 5p/2-3p

......

-5p/2

3

X[k]

k0 1 2-3 -2 -1

1

3

2

4

......

75 6 8 109 11-6 -5 -4-9 -8 -7




(only one period is shown) (only one period is shown)

e) DTFT over one period:

i. , :

ii.

, :

PROBLEM 4 Find the DTFT representation of the following periodic signals:

a)

b)

c)

d) see figure

a)

Then:

3

x[n]

n

1 20-3 -2 -1

1

2

34d)

3

X[k]

k0 1 2

1

3

2

4 5 6

3

X[k]

k0 1 2-3 -2 -1

1

3

2




Finally:

Over one period:

b)

Then:

Finally:

Over one period:

c)

Then:

Finally:

Over one period:

d) We first determine the DTFS:

Finally:

Over one period:




PROBLEM 5 Given the following periodic signal:

a) Determine the FT of this periodic signal (Hint: use the inspection method to get the FS). Sketch it. b) If we sample with TS as the sampling period, get the FT of the impulse-sampled signal . c) Sketch the FT of the sampled signal for TS = 2, 5, 8 seconds. d) What is the minimum sampling period TS that avoids aliasing in the frequency domain?

a)

Fourier Series:

Then:

Finally:

b)

c)

w

0

X(jw)p/j

p/8 p/4-p/4 -p/8

-p/j

w

0

X(jw)p/j

p/8 p/4-p/4 -p/8

-p/j

... ...

p-p p-p/4 p+p/4-p+p/4-p-p/4




d) From (a), we see that

.

The min. sampling period that prevents aliasing is 3.9 seconds. Note that results in aliasing.

PROBLEM 6 A signal has the following FT representation: a) The signal is then sampled by TS seconds. Sketch the FT representation of the sampled signal for

the following sampling periods: TS = 0.2, 0.4, 0.6 seconds. b) What is the condition on the sampling period TS so that the sampling does not introduce aliasing?

a)

X(jw)

-3.5p

w

3.5p2pp-p-2p

1

w

0

X(jw)p/j

p/8 p/4-p/4 -p/8

-p/j

... ...

2p/5-2p/5

2p/5-p/4

2p/5+p/4

-2p/5+p/4

-2p/5-p/4

w

0

X(jw)p/j

p/8 p/4-p/4 -p/8

-p/j

... ...

p/4+p/4

-p/4-p/4

X(jw)

-3.5p

w

3.5p

1

6.5p 13.5p10p-13.5p -6.5p-10p

......

X(jw)

-3.5p

w

3.5p

1

1.5p 8.5p5p-8.5p -1.5p-5p

......




b)

PROBLEM 7 A signal has the following FT representation: a) Sampling this signal will always introduce aliasing. For this reason, the signal is first passed through a

system called anti-aliasing filter, which bounds the FT of the signal (as shown in the figure below). For TS = 0.5, 0.25, 0.125 seconds, what is the minimum width (in rads/s) of the anti-aliasing filter so that no aliasing is introduced to the signal ?

b) Sketch the FT of the anti-aliasing filter (it should go from ). Also, provide the time-domain

impulse response of the anti-aliasing filter. c) Now, we want to recover after the sampling process. We need an ideal reconstruction filter.

Sketch the operation this filter applies on the FT of the sampled signal . Also provide the time-

domain impulse response of this ideal reconstruction filter (use a generic with

).

a)

X(jw)

w

2pp-p-2p

1

Ideal Reconstruction Filter

xAd(t) xA(t)

Anti-aliasingFilter

x(t)xA(t)

X(jw)

w

2pp-p-2p

1

XA(jw)

w

-wm

1

wm

ImpulseSampling

xAd(t)

X(jw)

-3.5p

w

3.5p

1

-0.166p

6.88p

3.33p

-6.88p 0.166p

-3.33p

......




b)

c)

1

w

HA(jw)

-wm wm

XAd(jw)

-wm

1/Ts

wm wS-wm-wS wS wS+wm-wS+wm-wS-wm

Ts

w

-wm wm

1

w

HR(jw)

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