ch1 (dspreview)-1

7/30/2019 ch1 (DSPreview)-1

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Signal are represented mathematically asfunctions of one or more independentvariables.

Digital signal processing deals with thetransformation of signal that are discrete inboth amplitude and time.

Discrete time signal are representedmathematically as sequence of numbers.

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A discrete time system is definedmathematically as a transformation oroperator.

y[n] = T{ x[n] }

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T{.}x [n] y [n]


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][][]}[{]}[{]}[][{ 212121 nynynxTnxTnxnxT

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The class of linear systems is defined by theprinciple of superposition.

And

Where a is the arbitrary constant.

The first property is called the additivity propertyand the second is called the homogeneity or scalingproperty.

][]}[{]}[{ naynxaTnaxT


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][1 nx

5

These two property can be combined into theprinciple of superposition,

]}[{]}[{]}[][{ 2121 nxbTnxaTnbxnaxT

][][ 21 nbxnax ][2 nx

H

H

Linear System

H

][][ 21 nbynay

][1 ny

][2 ny


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A Time-Invariant system is a system for witcha time shift or delay of the input sequencecause a corresponding shift in the output

sequence.

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][1 nxH

H

][ 01 nnx

][1 ny

][ 01 nny


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A particular important class of systems consistsof those that are linear and time invariant.

LTI systems can be completely characterized by

their impulse response.

From principle of superposition:

Property of TI:

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k

knkxTny ][][][

k knTkxny ][][][

k

knhkxny ][][][


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k

knhkxny ][][][

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Above equation commonly called convolutionsum and represented by the notation

][][][ nhnxny


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Commutativity:

Associativity:

Distributivity:

Time reversal:

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][][][][ nxnhnhnx

][][][ nhnxny

])[][(][][])[][( 321321 nhnhnhnhnhnh

])[][(])[][(])[][(][ 2121 nxnhbnxnhanbxnaxnh


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If two systems are cascaded,

The overall impulse response of the combined

system is the convolution of the individual IR:

The overall IR is independent of the order:

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H1 H2

H2 H1

][][][ 21 nhnhnh


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Infinite-duration impulse-response (IIR).

Finite-duration impulse-response (FIR)

In this case the IR can be read from the right-

hand side of:

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][...]1[][][ 10 qnxbnxbnxbny q

nbnh ][


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Transforms are a powerful tool for simplifyingthe analysis of signals and of linear systems.

Interesting transforms for us:

Linearity applies:

Convolution is replaced by simpler operation:

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][][][ ybTxaTbyaxT

][][][ yTxTyxT


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Most commonly transforms that used incommunications engineering are:

Laplace transforms (Continuous in Time & Frequency)

Continuous Fourier transforms (Continuous in Time)

Discrete Fourier transforms (Discrete in Time)

Z transforms (Discrete in Time & Frequency)

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Definition Equations:

Direct Z transform

The Region Of Convergence (ROC) plays an

essential role.

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n

nznxzX ][)(


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Elementary functions and their Z-transforms: Unit impulse:

Delayed unit impulse:

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][][ knnx

0:][)(

zROCzzknzX

n

kn

][][ nnx

0:1][)(

zROCznzX

n

n


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Unit Step:

Exponential:

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][][ nuanx n

||:1

1)(0

1azROC

azzazX

n

nn

otherwise0,

0n,1][nu

1:1

1)(

0 1

zROC

zzzX

n

n


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Important Z Transforms

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Region Of Convergence

(ROC)

Whole Page

Whole Page

Unit Circle

|z| > |a|


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Elementary properties of the Z transforms:

Linearity:

Convolution: if

,Then

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)()(][][ zbYzaXnbynax

][][][ nynxnw

)()()( zYzXzW


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Shifting:

Differences: Forward differences of a function,

Backward differences of a function,

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)(][ zXzknx k

][]1[][ nxnxnx

]1[][][ nxnxnx


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Since

the shifting theorem

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][]1[][][ nnnxnx

)()1(][ zXznxZ

)()1(][ 1 zXznxZ


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The ROC is a ring or disk in the z-planecentered at the origin :i.e.,

The Fourier transform of x[n] converges atabsolutely if and only if the ROC of the z-transform of x[n] includes the unit circle.

The ROC can not contain any poles.

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If x[n] is afinite-duration sequence, then the ROCis the entire z-plane, except possibly or.

If x[n] is a right-sided sequence, the ROC extendsoutward from the outermost finite pole into .

The ROC must be a connected region.

22

0z z

)(zX z


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A two-sidedsequence is an infinite-duration sequencethat is neither right sided nor left sided.

If x[n] is a two-sided sequence, the ROC will

consist of a ring in the z-plane, bounded on theinterior and exterior by a pole and not containingany poles.

If x[n] is a left-sided sequence, the ROC extends inward from the innermost nonzero pole in to

.

23

0z)(zX


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We have seen that By the convolution property of the Z transform

Where H(z) is the transfer function of system.

Stability

A system is stable if a bounded input produced abounded output, and a LTI system

is stable if:

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][][][ nhnxny

)()()( zHzXzY

Mnx |][|

k

kh |][|


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Fourier Transform

Fourier Series

DiscreteTime Continuous FFT

Discrete Time Discrete FFT

Time Frequency Transform Type

Continuous

Discrete

Continuous

Continuous

Continuous Discrete

Discrete Discrete


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Definition Equations:

Direct Z transform

It is customary to use the

Then the direct form is:

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1

0

2

][][N

n

N

knj

enxkX

N

j

N eW2

1

0

][][N

n

nkNWnxkX


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With the same notation the inverse DFT is

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1

0

][1

][N

k

nkWkX

Nnx


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Elementary functions and their DFT: Unit impulse:

Shifted unit impulse:

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][][ pnnx

kpWkX ][

][][ nnx

1][ kX


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Constant:

Complex exponential:

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njenx ][

2

][ NkNkX

1][ nx

][][ kNkX


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Cosine function:

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nfnx 02cos][

][][

2][

00

NfkNNfkN

kX


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Elementary properties of the DFT:

Symmetry: If

,Then

Linearity: if

and

,Then

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][][ kFnf

][][ kXnx

][][][][ kbYkaXnbynax

][][ nNFkf

][][ kYny


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Shifting: because of the cyclic nature of DFTdomains, shifting becomes a rotation.

if

,Then

Time reversal:

if

,Then

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][][ kXnx

][][ kXnx

][][ kXWpnx kp

][][ kXnx


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Cyclic convolution: convolution is a shift, multiplyand add operation. Since all shifts in the DFT arecircular, convolution is defined with this circularityincluded.

1

0

][][][][N

p

pnypxnynx

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