ch 1,2 - intro, systems, ft

8/3/2019 Ch 1,2 - Intro, Systems, FT

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Digital Signal Processing

Chapter 2: Discrete-time Signals and Systems

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Discrete-time signal processing

Sampling, no digitization

Digital Signal Processing Sampling, digitization

Signal:

Digital Signal Processing Any function of one or more

variables which contains useful information

Signals One dimensional

Speech signals

Multi-dimensional

Pictures

3


Discrete-time signals

May be discrete from the beginning

Could have been the result of sampling a continuous time

signal

Discrete-time

ProcessingD-to-AA-to-D

x(t) y(t)y[n]x[n]

Cont.-time

Signal

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Discrete-Time Sequences

Discrete-time sequence

Graphical representation of a discrete-time signal.

[ ]{ } ,nxx = ,


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Discrete-Time Sequences

Segment of a continuous-time speech signal.

Sequence of samples obtained with T=125 us

)(][ nTxnx d= ,


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Basic Sequences

Sinusoidal sequences

[ ],),cos(

0nallfornAnx +=

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Periodic Signals

)cos( 0n

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Periodic Signals

)cos( 0n

12

Basic Sequences

Exponential sequences

[ ] .nAnx =


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Basic Sequences

Exponential sequence where is complex

If

)( 0 += njneA

).sin()cos( 00 +++= nAjnAnn

[ ] njnjn eeAAnx 0 ==

jeAA =0 je=

[ ] )sin()cos( 00)( 0

+++== + nAjnAeAnx nj

1=

[ ] .nAnx =

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Basic Sequences

Complex exponential sequence

[ ]

nj

Aenx

)2( 0 +=

.002 njnjnj AeeAe

==

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Discrete-time Systems

Any operation that maps an input sequence to an output

sequence

[ ] [ ]}{ nxTny =

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Example: Ideal delay System

Example: Moving average system

[ ] [ ]


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Moving average

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Memoryless system

Example:

for each value of n.

[ ] [ ]( ) ,2nxny =

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Linear Systems

If

Then

Or we can combine the two conditions as

[ ] [ ]{ } [ ]{ } [ ]{ } [ ] [ ]

[ ]{ } [ ]{ } [ ],

212221

naynxaTnaxT

nynynxTnxTnxnxT

==

+=+=+

[ ] [ ]

[ ] [ ]nynx

nynx

T

T

22

11

[ ] [ ]{ } [ ]{ } [ ]{ }nxbTnxaTnbxnaxT 2121 +=+

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Linear Systems

Accumulator

Linear?

[ ] [ ]=

=n

k

kxny


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Time Invariance

If

Then

Example:

Time invariant?

[ ] [ ]nynxT

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[ ] [ ]0101 nnynnxT

[ ] [ ]=

=n

k

kxny

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Time Invariance

Example:

Time invariant?

LTI Systems: Linear and time-invariant

[ ] [ ] ,,


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LTI Systems

LTI Systems are completely characterized by their

impulse response.

Given the input signal, the output can be determined.

[ ] [ ]

=

=k

knkxTny ][

[ ] [ ] { } [ ]

=

=

==kk

knhkxknTkxny ][][

[ ] ][*][ nhnxny =

Convolution

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Convolution

1. Flip one sequence (say h[k]) around origin h[-k]

2. Shift the flipped sequence h[n-k]

3. Multiply by the other sequence and add

[ ] [ ] [ ]

=

=

==kk

knxkhknhkxny ][][

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Output of an LTI System

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Output of an LTI System


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Convolution

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Convolution Example

[ ] )3,2,1(

=nh

[ ] )1,1,1(=nx

[ ] ?][*][ == nhnxny

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Convolution Example

In general:

Thus

[ ] [ ] [ ]== Nnununh

.,0

,10,1

otherwise

Nn

[ ] [ ].nuanx n=

.10 Nnfor

[ ] [ ]

=

=

=

=

==

n

k

k

k

k

k

a

knhaknhkxny

0

0

][][

12

1

.1

2

1

21

NNan

Nk

NNk

=

=

+

[ ] .10,1

1 1

=

+

Nna

any

n

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Example


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Example (Cont.)

When n>N-1

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Properties of LTI Systems

These properties can be proved easily using the definition

of convolution operation.

Commutative property

Distributive property

[ ] [ ] [ ] [ ]nxnhnhnx =

[ ] [ ] [ ] [ ] [ ] [ ] [ ]nxnhmnxmhmhmnxnymm

===

=

=

[ ] [ ] [ ] [ ] [ ] [ ] [ ]nhnxnhnxnhnhnx 2121 )( +=+

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Serial combination of DT systems

h1[n] h2[n]

h2[n] h1[n]

h1[n]* h2[n]

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Parallel combination of DT systems


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LTI systems are BIBO stable if and only if the impulse

response is absolutely summable

[ ]


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Example

Causal and stable? FIR or IIR?

Ideal delay system:

Moving average system:

[ ] [ ]dnnxny =

[ ] [ ]dnnnh = nda positive fixed integer

[ ] [ ]= ++=2

11

121

M

Mk

knxMM

ny

[ ] [ ] =++

= =

2

11

1

21

M

MK

knMM

nh

++.,0

,,1

121

21

otherwise

MnMMM

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Example

Causal and stable? FIR or IIR?

Accumulator:

Forward Difference

Backward Difference

[ ] [ ]

=

=n

k

kxny

[ ] [ ]

=

=k

knh

0; a =0.9 (solid curve) and a=0.5 (dashed curve).

Magnitude

Phase

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Symmetry Property of Fourier Transform

Frequency response for a system with impulse response h[n] = an u[ n].

a > 0; a =0.9 (solid curve) and a=0.5 (dashed curve).

Real part

Imaginary part


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Existence of Fourier Transform

Does the infinite sum converge to a finite value?

If the sequence is absolute summable, its Fourier

transform exists. (Sufficient Condition)

( )

ch 1,2 - intro, systems, ft

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