Chapter 2_dsp1_co thuc_full

7/31/2019 Chapter 2_dsp1_co thuc_full

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CHAPTER 2:

DISCRETE-TIME SIGNALS & SYSTEMS

Lesson #4: DT signals

Lesson #5: DT systems

Lesson #6: DT convolution

Lesson #7: Difference equation models

Lesson #8: Block diagram for DT LTI systems

Duration: 9 hrs


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Lecture #4

DT signals

1. Representations of DT signals

2. Some elementary DT signals

3. Simple manipulations of DT signals

4. Characteristics of DT signals


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Converting a CT signal into a DT signal by sampling: given xa(t) to

be a CT signal, xa(nT) is the value ofxa(t) at t = nT DT signal is

defined only forn an integer

n),n(x)nT(x)t(x anTta

-2T -T 0 T 2T 3T 4T 5T 6T 7T . . . nT

t

Sampled signals


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n -1 0 1 2 3 4

x[n] 0 0 1 4 1 0

1.Functional representation

n,0

2n,4

3,1n,1

]n[x

Representations of DT signals

2.Tabular representation


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3.Sequence representation

Representations of DT signals

1,4,1,0][nx

-1 0 1 2 3 4 5 n

4.Graphical representation 4

1 1


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Lecture #4

DT signals




4. Classification of DT signals


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1. Unit step sequence

2. Unit impulse signal

3. Sinusoidal signal

4. Exponential signal

Some elementary DT signals


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1 0[ ]

0 0

nu n

n

Unit step

-1 0 1 2 3 4 5 6 n

1 1 1


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Time-shifted unit step

0

0

0nn,0

nn,1]nn[u

0 -n0-1 n0 n0+1 n

For n0 > 0

1 1 1


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Time-shifted unit step

0

0

0nn,0

nn,1]nn[u

-n0-1 n0 n0+1 0 n

For n0 < 0

1 1 1


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Unit impulse

1 0[ ]

0 0

nn

n

-2 -1 0 1 2 n

1


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Time-shifted unit impulse

0

0

0nn,0

nn,1]nn[

For n0 > 0

0 n0-1 n0 n0+1 n

1


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Time-shifted unit impulse

0

0

0nn,0

nn,1]nn[

n0-1 n0 n0+1 0 n

1


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Relation between unit step andunit impulse

]n[x]nn[]n[x

]nn[]n[x]nn[]n[x

]1n[u]n[u]n[

]k[]n[u

0

n

0

000

n

k


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Sinusoidal signal

n),nF2cos(A

n),ncos(A)n(x


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Exponential signal

nCa]n[x

1. IfCand aare real, then x[n]is a real exponential

a > 1 growing exponential

0 < a < 1 shrinking exponential

-1 < a < 0 alternate and decay

a < -1 alternate and grows

2. IfCor aor both is complex, then x[n]is a complexexponential


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An example of real exponential signal

nnx )2.1)(2.0(][


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An example of complexexponential signal

nj

enx 6121

2][


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Periodic exponential signal

Recall:A DT sinusoidal signal is periodic only if its

frequency is a rational number

Consider complex exponential signal:

It is also periodic only if its frequency is a rational number:

N

kor

N

kF

2

00

)sin()cos(][ 000 njnCCenx

nj


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Fundamental period

The fundamental period can be found as

Where k is the smallest integer such that N is an integer

Step 1: Is rational?

Step 2: If yes, then periodic; reduce to

0

2kN

spo

cycles

N

k

int#

#

2

0

2

0


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Examples

Determine which of the signals below are periodic. For the

ones that are, find the fundamental period and

fundamental frequency nj

enx6

1 ][

612

22

12

12

1

)2(62

0

0

N

N

N

k One cycle in 12 points

: fundamental period

: fundamental frequency


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Examples

Determine which of the signals below are periodic. For theones that are, find the fundamental period and

fundamental frequency

15

3

sin][2 nnx

510

22

10

10

3

)2(5

3

2

0

N

N

N

k 3 cycles in 10 points


: fundamental frequency 0because k 1


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Examples




)2cos(][3 nnx

0 = 2 0/2 = 1/ is irrational x3[n] is not periodic


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Examples




)2.1cos(][4 nnx

5

22

5

5

3

2

2.1

2

0

N

N

N

k 3 cycles in 5 points


: fundamental frequency 0because k 1


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Lecture #4

DT signals




4. Classification of DT signals


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Adding and subtracting signals

Transformation of time:

- Time shifting

- Time scaling

- Time reversal

Transformation of amplitude:

- Amplitude shifting

- Amplitude scaling

- Amplitude reversal

Simple manipulations of DT signals


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Do it point by point

Can do using a table, or graphically, or by

computer program

Example: x[n] = u[n] u[n-4]

Adding and Subtracting signals

n =4

x[n] 0 1 1 1 1 0


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x[n] x[n - k]; k is an integer

k > 0: right-shift x[n] by |k| samples

(delay of signal)

k < 0: left-shift x[n] by |k| samples

(advance of signal)

Time shifting a DT signal


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Examples of time shifting

-1 0 1 2 3 4 n

4

1 1

x[n]

-1 0 1 2 3 4 n

4

1 1

x[n-2]


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Examples of time shifting

-1 0 1 2 3 4 n

4

1 1

x[n]

-1 0 1 2 3 4 n

4

1 1

x[n+1]


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Time scaling a DT signal

x[n] y[n] = x[an]

|a| > 1: speed up by a factor of a

a must be an integer

|a| < 1: slow down by a factor of a

a = 1/K; K must be an integer


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Examples of time scaling

-2 -1 0 1 2 n

4

-1 0 1 2 n

1

x[2n]x[2n+1]

-1 0 1 2 3 4 n

4

1 1

x[n]


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Given x[n]

Examples of time scaling

w1[n] = x[2n]

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

1

2

2

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

What does w1[n/2]look like? Just look like x[n]!


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x[n] x[-n]

Flip a signal about the vertical axis

Time reversal a DT signal


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Examples of time reversal

-1 0 1 2 3 4 n

4

1 1

x[n]

-3 -2 -1 0 1 2 3 4 n

4

1

x[-n]


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

Method 1: Flip first, then shift

Method 2: Shift first, then flip

Combining time reversal and time shifting


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4


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Example

Method 2-1 0 1 2 3 4 5 n

1

4

x[n]

-4 -3 -2 -1 0 1 2 3 4 5 n

w[-m] = x[-(n+2)] = x[-n-2]

x[n+2] = w[m]

-2 -1 0 1 2 3 4 5 6 n

-2 -1 0 1 2 3 4 5 6 7 8 m


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x[n] y[n] = x[an-b]

Method 1: time scale then shift

Method 2: shift then time scale

Be careful!!! For some cases, method 1 or 2 doesnt work.To make sure, plug values into the table to check

Combining time shifting and time scaling


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4


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Example

Method 1

x[-2n] = w[n]

-4 -3 -2 -1 0 1 2 3 4 5 n

-1 0 1 2 3 4 5 n

1

4

x[n]

-4 -3 -2 -1 0 1 2 3 4 5 n

w[n-1] = x[-2(n-1)] = x[-2n+2]


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x[n] y[n] = x[an-b]

Method 2: shift then time scale

Ex. Find y[n] = x[2-2n]

y[n] = x[-2(n-1)]

x[n]

x[n-1] = w[m]

w[-2m] = x[-2(n-1)]

Example Method 2

Delayby 1

Timescaleby -2

4


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Example

Method 2 -1 0 1 2 3 4 5 n

1

x[n]

-4 -3 -2 -1 0 1 2 3 4 5 n

w[-2m] = x[-2(n-1)] = x[-2n+2]

x[n-1] = w[m]

-1 0 1 2 3 4 5 n

-2 -1 0 1 2 3 4 m


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Example

y[n] = x[2n-3]??

n x[n] y[n]-1 0 0

0 0 0

1 1 02 4 1

3 1 1

4 0 0

-1 0 1 2 3 4 n

4

1 1

x[n]

-1 0 1 2 3 4 n

1y[n]


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Find

Exercise

[ ] ( [ 1] [ 5])( [2 ])x n u n u n nu n

-1

21

-1

x[n]

0 1 2 3 4 5 n


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Lecture #4

DT signals




4. Characteristics of DT signals


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Symmetric (even) and anti-symmetric (odd)signals

Energy and power signals

Characteristics of DT signals


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A DT signal xe[n] is evenif

And the signal xo[n] is oddif

Any DT signal can be expressed as the sum of an even

signal and an odd signal:

Even and odd signals

Even [ ] [ ]e ex n x n

Odd [ ] [ ]o o

x n x n

12

[ ] ( [ ] [ ])ex n x n x n

12

[ ] ( [ ] [ ])ox n x n x n

[ ] [ ] [ ]e ox n x n x n


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How to find xe[n]and xo[n]from a given x[n]?

Step 1: find x[-n]

Step 2: find

Step 3: find

Even and odd signals

12

[ ] ( [ ] [ ])e

x n x n x n

12

[ ] ( [ ] [ ])o

x n x n x n

Example


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Given x[n]

Example

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8

1

2

Find x[-n]

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

1

2

Example


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Find x[n] + x[-n]

Example

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8

1

2

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

1

3/2

1/2

Find xe[n]


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Determine which of the signals below are energy signals?

Which are power signals?

Examples

(a) Unit step1 0[ ]0 0

nu nn

0

22 1][nn

nxE

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1lim1

12

1lim][

12

1lim

0

22

N

N

Nnx

NP

N

N

nN

N

NnN

Unit step is a power signal


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E l


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Determine which of the signals below are energy signals?

Which are power signals?

Examples

(c) ])4n[u]n[u(n4

cos]n[x

]3[2

2]1[

2

2][

0

304

cos][ nnn

otherwise

nnnx

242

421

22

221][

22

2

n

nxE

x[n] is an energy signal


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I t ti f DT t


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y[n] = y1[n] + y

2[n] = T

1{x[n]} + T

2{x[n]} = (T

1+ T

2){x[n]}

= T{x[n]}

y[n] = T{x[n]}: notation for the total system

Interconnection of DT systems

System1T1{ }

x[n] y[n]

System2T2{ }Parallel

connection


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y[n] = T2{y1[n]} = T2{T1{x[n]}} = T{x[n]}

y[n] = T{x[n]}: notation for the total system

Interconnection of DT systems

System1T1{ }

x[n] y[n]System2T2{ }

Cascade connection


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Lecture #5

DT systems

1. DT system

2. DT system properties


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Memory

Invertibility

Causality

Stability

Linearity

Time-invariance

DT system properties

M


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y[n0] = f(x[n0]) system is memoryless (static)

Otherwise, system has memory (dynamic), meaning that its

output depends on inputs rather than just at the time of the

output

Ex:

a) y[n] = x[n] + 5: is memoryless

b) y[n]=(n+5)x[n]: is memoryless

c) y[n]=x[n+5]: has memory

Memory


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Invertibility

A system is said to be invertible if distinct inputs

result in distinct outputs

Ex.: y[n] = |x[n]| is not invertible


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Invertibility

Ti[T(x[n])] = x[n]

T() Ti()

x[n] x[n]

System Inverse system

A system is said to be invertible if distinct inputs

result in distinct outputs

Examples for invertibility


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Examples for invertibility

Determine which of the systems below are invertible

a) Unit advance y[n] = x[n+1]

b) Accumulator

c) Rectifier y[n] = |x[n]|

n

k

kxny ][][

1

][]1[

n

kkxny y[n] y[n-1] = x[n] is inverse system

is invertible

is invertible

is not invertible

y[n-1] = x[n] is inverse system


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The output of a causal system (at each time) does not

depend on future inputs

All memoryless systems are causal

All causal systems can have memory or not

Causality


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Examples for causality

Determine which of the systems below are causal:

a) y[n] = x[-n]

b) y[n] = (n+1)x[n-1]

c) y[n] = x[(n-1)2]

d) y[n] = cos(w0n+x[n])

e) y[n] = 0.5y[n-1] + x[n-1]

non causal

non causal

causal

causal

causal


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Examples for stability

Determine which of the systems below are BIBO stable:

a) A unit delay system

b) An accumulator

c) y[n] = cos(x[n])

d) y[n] = ln(x[n])

e) y[n] =exp(x[n])

stable

stable

stable

unstable

unstable


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Scaling signals and adding them, then processing through the system

same asProcessing signals through system, then scaling and adding them

Linearity

If T(x1[n]) = y1[n] and T(x2[n]) = y2[n]

T(ax1[n] + bx2[n]) = ay1[n] + by2[n]


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If you time shift the input, get the same output, but with the

same time shift

The behavior of the system doesnt change with time

Time-invariance

If T(x[n]) = y[n]

then T(x[n-n0]) = y[n-n0]

E l f li it d ti


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Examples for linearity and time-invariance

Determine which of the systems below are linear, whichones are time-invariant

a) [ ] [ ]y n nx n

Linear

Not time-invariant

E l f li it d ti


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Examples for linearity and time-invariance

Determine which of the systems below are linear, wichones are time-invariant

b) ]n[x]n[y2

Non-linear

Time-invariant


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E l f DT t ti


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Example for DT system properties

Given the system below:

a) It is memoryless

][5.1

5.2

][

2

nxn

n

ny

b) It is invertible its inverse system is: ][5.2

5.1][

2

nyn

nnx

c) It is causal

E l f DT t ti


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0 100 200 300 400 500 600 700 800 900 10000

1

2

3

4

5

6

7

8

9


Given the system below: ][5.1

5.2

][

2

nxn

n

ny

c) It is stable

For |x[n]| M then |y[n]| 9M

2

5.1

5.2

n

n

E l f DT t ti


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Given the system below: ][5.1

5.2

][

2

nxn

n

ny

d) For n = 0 y[0] = 2.778x[0]

For n = -1 y[-1] = 9x[-1] time invariant

e) Superposition linear

][][

][5.1

5.2][

5.1

5.2

][][5.1

5.2][][

2211

2

2

21

2

1

2211

2

2211

nyanya

nxn

nanx

n

na

nxanxan

nnxanxa

Lecture #6


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Lecture #6

DT convolution

1. DT convolution formula

2. DT convolution properties

3. Computing the convolution sum

4. DT LTI properties from impulse response

Computing the response of DT LTI


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Method 1:based on the direct solution of the input-output equation

for the system

Method 2:

Decompose the input signal into a sum of elementary signals

Find the response of system

to each elementary signal

Add those responses to obtain

the total response of the system

to the given input signal

Computing the response of DT LTIsystems to arbitrary inputs

k

kk

kk

k

kk

nycnynx

nynx

nxcnx

][][][

][][

][][

DT convolution formula


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Convolution: an operation between the input signal to a

system and its impulse response, resulting in the output signal

CT systems: convolution of 2 signals involves integrating the

product of the 2 signals where one of signals is flipped and

shifted

DT systems: convolution of 2 signals involves summing theproduct of the 2 signals where one of signals is flipped and

shifted

DT convolution formula

I l t ti f DT i l


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We can describe any DT signal x[n] as:

Example:

Impulse representation of DT signals

[ ] [ ] [ ]k

x n x k n k

-1 0 1 2 3 n

x[n]

-1 0 1 2 n

x[0][n-0]

-1 0 1 2 n

x[1][n-1]

-1 0 1 2 n

x[2][n-2]+ +

Impulse response of DT systems


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Impulse response: the output results, in response to a unit impulse

Denotation: hk[n]: impulse response of a system, to an impulse at

time k

Impulse response of DT systems

Time-invariantDT system

Time-invariant

DT system

[n]

[n-k]

h[n]

h[n-k]

Remember:the impulse response is a sequence of values that may

go on forever!!!

Response of LTI DT systems to


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p yarbitrary inputs

LTI DT system[n-k] h[n-k]

LTI DT system

[ ] [ ] [ ]k

x n x k n kk

knhkxny ][][][

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

Convolution sum

Convolution sum in more details


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Convolution sum in more details

y[0] = x[0] * h[0]

= + x[-2]h[2] + x[-1]h[1] + x[0]h[0]

+ x[1]h[-1] + x[2]h[-2] + +

The general output:

y[n] = + x[-2]h[n+2] + x[-1]h[n+1] + x[0]h[n]

+ x[1]h[n-1] + x[2]h[n-2] + + x[n-1]h[1]

+ x[n]h[0] + x[n+1]h[-1] + x[n+2]h[-2]

Note:the sum of the arguments in each term is always n

Lecture #6


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Lecture #6

DT convolution






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

[n-m] * x[n] = x[n-m]

[n] * x[n-m] = x[n-m]

Commutative law

Associative law

Distributive law

Convolution sum properties

Commutative law


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Commutative law

][*][][*][ nxnhnhnx

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

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

Associative law


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Associative law

])[*][(*][][*])[*][( 2121 nhnhnxnhnhnx

h1[n]x[n] y[n]

h2[n]

h2[n]x[n] y[n]

h1[n]

h1[n]*h2[n]

x[n] y[n]

Distributive law


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Distributive law

])[*][(])[*][(])[][(*][ 2121 nhnxnhnxnhnhnx

h1[n] + h2[n]x[n] y[n]

h1[n]

x[n] y[n]

h2[n]

Lecture #6


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Lecture #6

DT convolution





Computing the convolution sum


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1. Foldh[k]about k = 0, to obtain h[-k]

2. Shifth[-k]by n0to the right (left) ifn0is positive (negative), toobtain h[n0-k]

3. Multiplyx[k]and h[n0-k]for all k, to obtain the product

x[k].h[n0-k]

4. Sum up the product for all k, to obtain y[n0]Repeat from 2-4 fof all of n

kkknhkxnyknhkxny ][][][][][][ 00

The length of the convolution sum result


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The length of the convolution sum result

Suppose:

Length of x[k] is Nx N1 k N1 + Nx 1

Length of h[n-k] is Nh N2 n-k N2 + Nh 1

N1 + N2 n N1 + N2 + Nx + Nh 2

Length of y[n]:

Ny = Nx + Nh 1

[ ] [ ] [ ] [ ] [ ]k

y n x n h n x k h n k

Example 1


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Example 1

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

[ ] [ 1] [ 3] [ ]x n u n u n n [ ] 2 [ ] [ 3]h n u n u n

n

n

x[n]

h[n]

-1 0 1 2 3

-1 0 1 2 3

Ex1 (cont)


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h[-k]

h[k]

x[k]

-1 0 1 2 3 k

-1 0 1 2 3 k

-2 -1 0 1 ky[0] = 6;

Ex1 (cont)


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h[-k]

x[k]

-1 0 1 2 3 k

-2 -1 0 1 k

-4 -3 -2 -1 0 k

h[-1-k]

y[-1] = 2;


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Ex1 (cont)


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( )

y[2] = 8;

h[-k]

x[k]

-1 0 1 2 3 k

-2 -1 0 1 k

h[2-k]

-2 -1 0 1 2 k

Ex1 (cont)


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( )

y[3] = 4;

h[-k]

x[k]

-1 0 1 2 3 k

-2 -1 0 1 k

h[3-k]

-2 -1 0 1 2 k


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Example 2


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

x n a u n [ ] [ ]h n u n

Try it both ways (first flip x[n] and do the convolution and then flip

h[n] and do the convolution). Which method do you prefer?

1||

1||1

0

0 aif

aifa

a

a

n

nn

n

a

aaa

nnn

n

nn

n

1

1)1( 01

0

1

0

Remember!


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Example 3


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Find y[n] = x[n]*h[n] where x[n] = bnu[n] and h[n] = anu[n+2]

|a| < 1, |b| < 1, a b

b

a

ba

b

ab

b

ababnyn

nyn

n

nn

k

k

nkn

k

kn

1

1

][2

0][:2

3

2

22

]2[1

][1

1122

nuab

babany

nn

Therefore

Factor out bn toget form you know

Example 4


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Compute output of a system with impulse response

h[n] = an

u[n-2], |a| < 1when the input is x[n] = u[-n]

a

aanyn

a

aann

n

nk

k

k

k

1][:2

1

][:22

2

]3[1

]2[1

][2

nua

anu

a

any

nTherefore

Flipping x[n] because it is simpler

A constant Varies with n

Lecture #6


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DT convolution






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Calculation of the impulse response


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Applying the unit impulse function to the input-output equation

Ex.: ]1n[x2

1]n[x21]1n[y

41]n[y

]1[)8/5()4/1(][)2/1(][ 1 nunny n

Suppose this

system is causal

2

1

0

4

1

8

5

8

5

4

1

4

1]2[

2

1]3[

2

1]2[

4

1]3[

4

1

8

5]1[

2

1]2[

2

1]1[

4

1]2[

41

85

85

21

21

41]0[

21]1[

21]0[

41]1[

2

1]1[

2

1]0[

2

1]1[

4

1]0[

xxhh

hh

xhh

hh


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Examples


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a p es

1. Is h[n] = 0.5nu[n] BIBO stable? Causal?

2. Is h[n] = 3nu[n] BIBO stable? Causal?

3. Is h[n] = 3nu[-n] BIBO stable? Causal?

Stable

Not stable

Stable

Causal

Causal

Not causal


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Linear constant coefficient difference equations


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General form:

q

][...]1[][][...]1[][ 101 MnxbnxbnxbNnyanyany MN

1a,]rn[xb]kn[ya 0

M

0r

r

N

0k

k

N, M: non-negative integers

N: order of equation

ak, br: real constant coefficients


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Components for block diagram


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EX: an averaging system y[n] = 0.5(x[n] + x[n-1])


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Direct form II realization of a system


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-a1

Z-1

-a2

Z-1

-aN

b0

b1

b2

bN

x[n] y[n]

Suppose

M = N


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HW


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Prob.1

a) 9.1a (iii)

b) 9.1a (iv)

c) 9.2a (i)

d) 9.2a (ii)

e) 9.2a (vi)

f) 9.5a

Prob.2

a) 9.13a (iii)

b) 9.15d

c) 9.17a

d) 9.17b


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Chapter 2_dsp1_co thuc_full

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