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EE 451 Fall 2001 Homework 9: Signal Processing and Filter Design, Assignments of Digital Signal Processing

New Mexico Institute of Mining & Technology (NMT)Digital Signal Processing

Information for homework 9 in ee 451 (fall 2001) course. It includes various signal processing and filter design problems. Students are required to determine the minimum sampling frequency for no aliasing, sketch fourier transforms, design butterworth filters, and more. Essential for university students specializing in electrical engineering or signal processing.

Typology: Assignments

Pre 2010

Uploaded on 08/08/2009

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EE 451 Fall 2001
EE 451
Homework #9
Due October 22, 2001
1. A signal
๎˜€๎˜‚๎˜๎˜„๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡
is processed by the system shown in (a) below. The spectrum of
๎˜€๎˜‚๎˜๎˜‰๎˜ƒ๎˜†๎˜…๎˜Š๎˜‡
is shown in (b)
below, where
๎˜‹๎˜๎˜Œ๎˜๎˜Ž๎˜‘๎˜๎˜“๎˜’
๎˜ƒ๎˜•๎˜”๎˜—๎˜–๎˜˜๎˜–๎˜˜๎˜–๎˜‰๎˜‡
rad/sec. The discrete-time system
๎˜™
๎˜ƒ๎˜›๎˜š๎˜„๎˜‡
in an ideal lowpass filter with
frequency response
๎˜™
๎˜ƒ๎˜๎˜œ๎˜Ÿ๎˜ž๎˜ˆ !๎˜‡
๎˜Ž
"
๎˜”๎˜“#%$ &๎˜$๎˜„'(&*)+#
๎˜–,#.-/๎˜…๎˜ˆ01๎˜œ32/4๎˜576/๎˜œ
โ„ฆโˆ’โ„ฆ
1
X (j )โ„ฆ
a
0
0โ„ฆ
0
x (t) y (t)
C/D D/C
Discreteโˆ’Time
System
H(z)
(a)
(b)
aa
(a) What is the minimum sampling frequency
8:9;๎˜Ž
๎˜”=<=>
such that no aliasing occurs in sampling
the input?
(b) If
&:)
๎˜Ž?๎˜’
<
๎˜
, what is the minimum sampling frequency such that
@
๎˜๎˜‰๎˜ƒ๎˜†๎˜…๎˜Š๎˜‡
๎˜Ž
๎˜€๎˜‚๎˜๎˜„๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡
?
2. A continuous-time signal
๎˜€๎˜‚๎˜A๎˜ƒ๎˜†๎˜…๎˜Š๎˜‡
, with Fourier transform
B
๎˜A๎˜ƒDC
๎˜‹
๎˜‡
shown below, is sampled with a
sampling period
>
๎˜ŽE๎˜๎˜“๎˜’
<
๎˜‹๎˜F
to form the sequence
๎˜€:G H!I
๎˜Ž
๎˜€J๎˜๎˜‰๎˜ƒ๎˜†H!>K๎˜‡
.
โˆ’โ„ฆ0โ„ฆ0
1
X (j )โ„ฆ
a
0โ„ฆ
(a) Sketch the Fourier transform
B
๎˜ƒ๎˜๎˜œ
๎˜ž๎˜Š
๎˜‡
for
$ &๎˜$L'
๎˜’
.
(b) The signal
๎˜€MG H!I
is to be transmitted across a digital channel. At the receiver, the original signal
๎˜€๎˜‚๎˜๎˜„๎˜ƒ๎˜†๎˜…๎˜ˆ๎˜‡
must be recovered. Draw a block diagram of the recovery system and specify its charac-
teristics. Assume ideal filters are available.
(c) In terms of
๎˜‹F
, for what range of values of
>
can
๎˜€๎˜๎˜ƒ๎˜†๎˜…๎˜Š๎˜‡
be recovered from
๎˜€MG H!I
?
1
pf3

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Download EE 451 Fall 2001 Homework 9: Signal Processing and Filter Design and more Assignments Digital Signal Processing in PDF only on Docsity!

EE 451

Homework #

Due October 22, 2001

  1. A signal

is processed by the system shown in (a) below. The spectrum of

is shown in (b)

below, where 

rad/sec. The discrete-time system 

in an ideal lowpass filter with

frequency response

โˆ’ฮฉ ฮฉ

1

X (j ฮฉ )

a

0

0

ฮฉ

0

x (t)

y (t)

C/D D/C

Discreteโˆ’Time

System

H(z)

(a)

(b)

a

a

(a) What is the minimum sampling frequency 8 :9;

such that no aliasing occurs in sampling

the input?

(b) If

, what is the minimum sampling frequency such that @

?

  1. A continuous-time signal

A 

, with Fourier transform B

ADC 

shown below, is sampled with a

sampling period

E

F to form the sequence

:G H!I

J H!>K

.

โˆ’ฮฉ

0

ฮฉ

0

1

X (j ฮฉ )

a

0

ฮฉ

(a) Sketch the Fourier transform B

for

$L'

.

(b) The signal

MG H!I

is to be transmitted across a digital channel. At the receiver, the original signal

must be recovered. Draw a block diagram of the recovery system and specify its charac-

teristics. Assume ideal filters are available.

(c) In terms of F, for what range of values of

can

be recovered from

MG H!I

?

  1. In the system below, B

ADC 

and 

are as shown. Note that FNOQP;RSP

T

. Sketch and

label the Fourier transform of @

for each of the following cases:

(a)

=<=>*U

=<=>!V

W

(b)

=<=>*U

=<=>!V

XYP

 W

(c)

=<=>*U

EZP

 W

,

=<=>[V

 W

(d)

=<=>*U

W

,

=<=>!V

EZP

W

X (j ฮฉ )

a

x (t)

y (t) C/D D/C

Discreteโˆ’Time

System

a H(z)

a

โˆ’ฮฉ ฮฉ

1

0

0

ฮฉ

0

j

1

H(e )

ฯ‰

ฯ‰

โˆ’ฯ€ โˆ’ฯ€/2 ฯ€/2 ฯ€

  1. A Butterworth analog lowpass filter is to be designed with the following specifications:

8 :]E^,

_

]

,`

R

dB

8 M9aEbc^,

_

9 aEd

dB

(a) What order filter e is required?

(b) What 3-dB frequency

will you use?

(c) Find the location of the poles of the filter in the

-domain.

(d) Write down the transfer function 

of the filter. Use second-order sections in the denominator

so all coefficients are real.

(e) Plot the frequency response of the filter using the MATLAB freqs function. Verify that the

filter meets the specifications.

(f) Check your answers to (a), (b) and (c) using the MATLAB buttord and butter functions.

  1. A Butterworth analog high-pass filter is to be designed with the following specifications:

8 :]Eb^,

_

]

,`

R

dB

Ec^,

_

Ed

dB

(a) Using gfN

find the specifications of the lowpass transfer function ih \

which can be used

as a prototype to design the highpass filter.

(b) What order filter e is required?

(c) What 3-dB frequency

will you use?