2b manipulation of signal

7/28/2019 2b Manipulation of Signal

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Manipulate

Sampled Signal

Lecture 2a


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%Graph 1

%Sawtooth sequence

n = 0:20;

y = sawtooth((n*pi)/4,1);

stem(n,y);

figure;

y1 = sawtooth((n*pi)/4,0.5);

stem(n,y1);

%Graph 2

%Square sequence

figure;

n = 0:20;

y = square((n*pi)/4,25);

stem(n,y);

figure;

y = square((n*pi)/4,75);

stem(n,y);

Type the following Matlab commands generate the following graphs:

PS: Type help function_name for brief description of a specific function.

0 2 4 6 8 10 1 2 1 4 1 6 1 8 20-1

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0.2

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0 2 4 6 8 10 1 2 1 4 16 1 8 2 0-1

-0.8

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0 2 4 6 8 10 1 2 1 4 16 1 8 2 0-1

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0 2 4 6 8 10 1 2 1 4 16 1 8 20-1

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1


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Operations on Sequences

A single-input, single-output discrete-time system operates on a

sequence, called the input sequence, according to some prescribed

rules and develop another sequence, called the output sequence,

usually with more desirable properties. Example : to remove

noise from an corrupted input signal.

Basic operations :

x[n]

y[n]

w1[n] +x[n]

y[n]

w2[n]

w1[n] =x[n]y[n]modulation / product

w2[n] =x[n] +y[n]addition

x[n] w3[n] x[n] w4[n]A D

x[n] w5[n]D-1

Time reversal : w6 =x[-n], a time-reversed version ofx[-n].

N-delay : w7 =x[n-N], ifNis positive.

IfNis negative it becomes an advance of the sequence

w3[n] =Ax[n]

multiplication

w4[n] =x[n - 1]

unit delay

w5[n] =x[n + 1]

unit advance


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Modulation examples :

To transform a sequence with low-frequency sinusoidalcomponents to a sequence with high-frequency components.

Example :

x[n] = { -1.1, 2.0, 7.4, -3.6, 0.1 }, y[n] = { 4.1, 0.8, -1.0, 4.2, 3.7 }

w1

[n] = {-4.51, 1.60, -7.40, -15.12, 0.37}

w2[n] = {3.00, 2.80, 6.40, 0.60, 3.80}

w3[n] = {-4.95, 9.00, 33.30, -16.20, 0.45} whereA = 4.5

w4[n] = {*, -1.10, 2.00, 7.40, -3.60, 0.10}

w5[n] = {-1.10, 2.00, 7.40, -3.60, 0.10 }

w6[n] = {0.10, -3.60, 7.40, 2.00, -1.10}

w7[n] = {*, *, *, -1.10, 2.00, 7.40, -3.60, 0.10 } whereN= 3.U1f19.m

Example :

x[n] = {-1.1, 2.0, 7.4}, y[n] = {4.1, 0.8, -1.0, 4.2, 3.7}

w1[n] =x[n] y[n] = {-4.51, 1.60, -7.40, 0, 0}

w2[n] =x[n] + y[n] = {3.00, 2.80, 6.40, 4.20, 3.70}

Note thatx[n] has been appended with two zeros to make it as

sequence of length of 5 in order to perform the above operations.

D D D

1 2 3 4

+

y[n]

x[n-1] x[n-2] x[n-3]x[n]

Example :

What isy[n], the output of the

discrete system, in terms ofx[n] ?

y[n] = 1x[n] + 2x[n-1] +

3x[n-2] + 4x[n-3]


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What isy[n], the output of the discrete system ?

y[n] = b0x[n] + b1x[n-1] + b2x[n-2] + a0y[n-1] + a1y[n-2]

y[n]+

b0

D

D

b1

b2

D

D

a0

a1

y[n-1]

y[n-2]

x[n-1]

x[n-2]

x[n]

System Impulse Response

[n] h[n]

?

x[n]

y[n] =x[n] h[n]

Convolution:

n-1 1 2 3 4 5 6 7

1

-2-3 n-1 1 2 3 4 5 6 7

1

-2-3

23

n-1 1 2 3 5 6 7-2-3


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An LTI discrete-time system is completely defined by its impulse

response i.e. knowing the impulse response, we can compute theoutput of the system to any arbitrary input.

Example : Let h[n] denotes the impulse response of the LTI

discrete-time system of interest. Let the input be

x[n] = 0.5 [n + 2] + 1.5 [n - 1] - [n - 2] + [n - 4] + 0.75 [n - 6]

Since the system is time-invariant, its responses to

0.5[n + 2], 1.5[n - 1], [n - 2], [n - 4] and 0.75[n - 6] will be

0.5h[n + 2], 1.5h[n - 1], h[n - 2], h[n - 4] and 0.75h[n - 6]

respectively. Therefore, the output will bey[n] = 0.5 h[n + 2] + 1.5 h[n - 1] - h[n - 2] + h[n - 4] + 0.75 h[n - 6]

Input-Output Relationship

We can generalise the above result for an arbitrary input sequence

x[n] that can be expressed as

[ ] [ ] [ ]

=

=

k

knhkxny

or alternatively, [ ] [ ] [ ]

=

=

k

khknxny

A LTI system is completely characterized by its impulse response

h[n] in the sense that, given h[n], it is possible to compute the

outputy[n] due to any inputx[n].


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The above two equations are called the convolution sum of the

sequencesx[n] and h[n].

It can be represented compactly as :

y[n] =x[n] * h[n]

where * denotes the convolution sum.

Example : Let the input and impulse response of a LTI discrete-

time system bex[n] = [n + 1] + 2 [n] + 3 [n - 1] - [n - 2] and

h[n] = [n] - [n - 1] + 2 [n - 2]. Determine the output,y[n].

ConvolutionFormula:

Methods for Convolution Graphical / sliding method

Frequency domain evaluation

Matlab: y = conv(x, h)

Analytical evaluation

Graphical Convolution


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Convolution by Graphical Method

Convolution by Graphical Method (cont)


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x[n] = u[n] - u[n-6]

h[n] = e-n u[n]

y[n] =x[k] h[n - k]

y[0] =x[k] h[0 - k]

y[0] = . + 0 h[-5] + 0 h[-4] + 0 h[-3] + 0 h[-2] + 0 h[-1] + 1 . 1 +x[1] 0 +x[2] 0 +x[3] 0 +x[4] 0 +

x[5] 0 + . = 1

y[1] =x[k] h[1 - k]

y[0] = . + 0 h[-5] + 0 h[-4] + 0 h[-3] + 0 h[-2] + 0 h[-1] + 1 . e-1 + 1 1 +x[2] 0 +x[3] 0 +x[4] 0 +

x[5] 0 + . = 1.3679..

Graphical Convolution

[1]


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Convolution by Sliding Method

For simple sequences it is possible to determine the output of

convolution by reversing one of the two convolving sequences andslide it against the other fixed sequence.

The sum of the resultant sequence for every slide step, n is

equivalent toy[n].

See the diagrams in the next pages for the sliding technique.

0 0 0 0 0 0 0 0 1 2 3 -1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 -1 2 0 0 0 0 0 0 0

x[k]

h[k] =

0 0 0 0 0 2 -1 1 0 0 0 0 0 0 0 0 0 0 0h[-2-k]

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0y[-2] =

=

0 0 0 0 0 0 0 0 1 2 3 -1 0 0 0 0 0 0 0x[k]

0 0 0 0 0 0 2 -1 1 0 0 0 0 0 0 0 0 0 0h[-1-k]

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0y[-1] =

=

= 0

= 1

0 0 0 0 0 2 -1 1 0 0 0 0 0 0 0 0 0 0 0h[-2-k]

0 0 0 0 0 0 0 0 1 2 3 -1 0 0 0 0 0 0 0x[k] =

k= 0

Let

Let


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0 0 0 0 0 0 0 0 1 2 3 -1 0 0 0 0 0 0 0x[k]

0 0 0 0 0 0 0 2 -1 1 0 0 0 0 0 0 0 0 0h[-k]

0 0 0 0 0 0 0 0 -1 2 0 0 0 0 0 0 0 0 0y[0] =

=

= 1

0 0 0 0 0 0 0 0 1 2 3 -1 0 0 0 0 0 0 0x[k]

0 0 0 0 0 0 0 0 2 -1 1 0 0 0 0 0 0 0 0h[1-k]

0 0 0 0 0 0 0 0 2 -2 3 0 0 0 0 0 0 0 0y[1] =

=

= 3

0 0 0 0 0 0 0 0 1 2 3 -1 0 0 0 0 0 0 0x[k]

0 0 0 0 0 0 0 0 0 2 -1 1 0 0 0 0 0 0 0h[2-k]

0 0 0 0 0 0 0 0 0 4 -3 -1 0 0 0 0 0 0 0y[2] =

=

= 0

0 0 0 0 0 0 0 0 1 2 3 -1 0 0 0 0 0 0 0x[k]

0 0 0 0 0 0 0 0 0 0 2 -1 1 0 0 0 0 0 0h[3-k]

0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 0y[4] =

=

= 7

0 0 0 0 0 0 0 0 1 2 3 -1 0 0 0 0 0 0 0x[k]

0 0 0 0 0 0 0 0 0 0 0 2 -1 1 0 0 0 0 0h[4-k]

0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0y[5] =

=

= -2

0 0 0 0 0 0 0 0 1 1 3 0 7 -2 0 0 0 0 0y[n]

n (+ve)

[4]

[3]


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Matlab provides a built-in function called conv which can be used to

compute the convolution between two finite-length sequences. It

assumes that the two sequences begin at n = 0.

Example (2-15) : Find the convolution outputy[n] betweenx[n] = [

3 11 7 0 -1 4 2 ] and h[n] = [ 2 3 0 -5 2 1].

x = [ 3 11 7 0 -1 4 2];

h = [ 2 3 0 -5 2 1];

y = conv(x,h)

y =

6 31 47 6 -51 -5 41 18 -22 -3 8 2

The convolution between two sequences of lengthsNandMalwaysresults in an output sequence of lengthN+M- 1.

Convolution in Matlab

However, conv does not provide or accept any timing information.Example : Find the convolution outputy[n] betweenx[n] = [ 3 11

7 0 -1 4 2 ] for -3 n 3 and h[n] = [ 2 3 0 -5 2 1] for -1 n

4.

The sequence generated in the previous example can be used with

additional timing information need to be determined as follows :

Starting point fory[n] : ny-s = -3 + -1 = -4.

End point fory[n] : ny-e = 3 + 4 = 7

y =

6 31 47 6 -51 -5 41 18 -22 -3 8 2

n = 0


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Example : Determine the outputy[n] for an LTI system with input

ofx[n] = u[n] - u[n-10] and h[n] = 0.9n u[n].

To generatey[n] using Matlab we need to define finite-lengthsequences. Assume thatx[n] and h[n] are defined for 0 n < 30

0 5 10 15 20 25

0

0.5

1

n

amplitude

0 5 10 15 20 25

0

0.5

1

n

amplitude

0 10 20 30 40 50

0

2

4

6

n

amplitude

Starting point is 0 while end

point is 29+29 = 58.

U2f7.m

x[n]

h[n]

y[n]

Example : Determine the response of a system with

h[n] = - u[n] + u[n-3] + 2u[n-4] and input,x[n] = [ 0 0 -2 -1 2 3 0

0 ] for 2 n 5.

-2 0 2 4

-2

0

2

4

n

amplitude

-5 0 5 10

-2

-1

0

1

2

3

n

amplitude

-5 0 5 10 15

-10

-5

0

5

10

n

amplitude

U2f7a.m

x[n]

h[n]

y[n]


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function [y,ny] = conv_m(x,nx,h,nh)

%Modified convolution routine for signal processing

%--------------------------------------------------

% y = convolution result

% ny = time index for y

% x = input vector

% nx = time index for x

% h = impulse response

% nh = time index for h

nyb = nx(1)+nh(1); nye = nx(length(x)) + nh(length(h));

ny = [nyb:nye];

y = conv(x,h);

A new function conv_mhas been defined and used here :

The function take into account the timing information of the input

sequences expressed in the form of a matrix : [ starting value ofn :

ending value ofn ]. Output sequence is returned together with

timing information matrix.

- Assume thatx[n], h[n] andz[n] are sequences

Commutativity

x[n] *y[n] =y[n] *x[n]

Associativity

x[n] * (y[n] *z[n] ) = (x[n] *y[n] ) *z[n]

Distributivity

x[n] * (y[n] +z[n] ) =x[n] *y[n] +x[n] *z[n]

The identity sequence for convolution operator is [n]

x[n] * [n] = [n] *x[n] =x[n]

The convolution of delayed unit sample sequence withx[n]

x[n] * [n - k] =x[n k]

Properties of Convolution


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There are two widely used schemes for developing complex LTI

discrete-time systems from simple LTI discrete-time systems.

h1[n] h2[n] h2[n] h1[n]

h1[n] * h2[n]

The overall impulse response h[n] of the cascade of two filters of

impulse responses h1[n] and h2[n] is given by

h[n] = h1[n] * h2[n]

Simple Interconnection Schemes

Cascade Connection

The cascade scheme is employed in the development of an inverse

system. If two LTI systems of impulse responses h1[n] and h2[n]

related to each other as follows :

h1[n] * h2[n] = [n]

then the two systems are said to be inverse of each other.

Parallel Connection

h1[n]

h2[n]

h1[n] + h2[n]+


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Correlation is an operation used in many applications in DSP. It is

a measure of the degree to which two sequences are similar.

Given two real-valued sequencesx[n] andy[n] of finite energy,

the crosscorrelation ofx[n] andy[n] is a sequence given by

The index l is called the shift or lag parameter. Wheny[n] =x[n],

we are looking at autocorrelation :

If we compare above with the convolution process, we can

conclude that : rx,y[l] =y[l] *x[-l] =y[-l] *x[l], crosscorrelationand rx,x[l] =x[l] *x[-l], autocorrelation. Therefore, conv can be

used to compute correlation.

[ ] [ ] [ ]

=

=

k

yx lkykxlr ,

[ ] [ ] [ ]

=

=

k

xx lkxkxlr ,

Correlation Process

Example : Recognition of signals (e.g. in radar systems). Let the

original signal sequence bex[n] = { 3, 11, 7, 0, -1, 4, 2 } and it is

corrupted by normally distributed noise sequence, w[n] of mean 0

and standard deviation of 1 :y[n] =x[n - 2] + w[n]. We will use

correlation process to recognise the original signal and its delay.

-2 0 2

-5

0

5

10

15

n

amplitude

0 2 4

-5

0

5

10

15

n

amplitude

-4 -2 0 2 4 6 8

-50

0

50

100

150

200

250

l

amplitude

It is clear that the correlation

exhibits a maximum value of

205 at l = 2. This implies

thaty[n] is similar tox[n]

shifted by 2.

x[n] y[n]


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Note:Convolution need flip of input /system .

But Correlation no flip of input / system

2b manipulation of signal

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