DIGITAL SIGNAL ROCESSING LECTURE 3 - University of...

DIGITAL SIGNAL PROCESSINGLECTURE 3Fall 20102K8-5th SemesterTahir MuhammadTahir [email protected]

Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and

Buck, ©1999-2000 Prentice Hall Inc.

Introduction to LTI System Impulse and Step Response

If the input to the DTS system is Unit Impulse p y p(δ[n]), then output of the system will be

Impulse Response (h[n]).

If the input to the DTS system is Unit Step (µ[n]), then output of the system will be

S R ( [ ])Step Response (s[n]).

Digital Signal Processing 2Lecture 3

Impulse Response


Input-Output Relationship

A Linear time-invariant system satisfied A Linear time invariant system satisfied both the linearity and time invariance properties.An LTI discrete-time system is characterized by its impulse responseExample:x[n] = 0.5δ[n+2] + 1.5δ[n-1] - δ[n-4]

will result iny[n] = 0.5h[n+2] + 1.5h[n-1] - h[n-4]


Input-Output Relationshipx[n] can be expressed in the form[ ] p

∑∞

−= knkxnx ][][][ δ

where x[k] denotes the kth sample of sequence {x[n]}The response to the LTI system is

−∞=k

t d

∑∑∞

−∞=

∞

−∞=

−=−=kk

khknxknhkxny ][][][][][or represented as

][][][ nhnxny ∗=Digital Signal Processing 5Lecture 3

][][][y

Computation of Discrete Convolution


Example 2.13

⎨⎧ −≤≤

=−−=101

][][][Nn

Nnununh][][ nunx nα=

⎩⎨ otherwise0

][][][ Nnununh

0for i) <n 1for iii) −> Nn

kn)1( −− Nn kn)1( −− Nn0

0][ =ny

10for ii) −≤≤ Nn⎞

⎜⎛ −

−−

==++−

−−=∑

a

aaaany

NN

nNnn

Nnk

k

1

1][

1

11

)1(

kn)1( −− Nn

⎟⎠

⎞⎜⎜⎝

⎛−

= +−

aaa Nn

111


aaanynn

k

k

−−

==+

=∑ 1

1][1

0

Finite Length Example

⎨⎧ ≤≤ 100 n

hnα⎧ ≤≤ 501 n

⎩⎨⎧ ≤≤

=otherwise0

100][

nnh

α

⎩⎨⎧ ≤≤

=otherwise0

501][

nnx

1x[n]

0 2

0.4

0.6

0.8

1

0.8

1h[n]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.2

0.4

0.6


0 1 2 3 4 5 6 7 8 9 100

n<0

0.8

1

x[n-k]h[n]

0 2

0.4

0.6

Digital Signal Processing Lecture 3 9-10 -8 -6 -4 -2 0 2 4 6 8 100

0.2

nn-5

0<= n <= 5

0.8

1

0.4

0.6

-2 0 2 4 6 8 10 120

0.2

nn-5

αα −==

+

∑ 11][

1nnkny

α−=∑ 10k

Digital Signal Processing Lecture 3 10

1

6<= n <= 10

0.6

0.8

1

0.2

0.4

-2 0 2 4 6 8 10 120

n-5 n

∑∑=

−+

−=

==5

0

)5(

5

][k

nkn

nk

kny αα05 knk

αααα −== −− ∑ 1

1][6

55

5 nknny


α−=∑ 10k

y

11<= n <= 15

0.8

1

0.4

0.6

-2 0 2 4 6 8 10 120

0.2

n-5 n

∑∑−

−+==n

nkkny15

)5(10

][ αα ∑∑=−= knk 05

αααα −==

−−

−− ∑ 1][

165

155

nn

nknny


α−=∑ 1

][0k

y

n > 15

0.8

1

0.4

0.6

-2 0 2 4 6 8 10 12 14 16 180

0.2

5 nn-5 n

0][ =ny


Input-output RelationshipProperties of convolutionProperties of convolution

Commutative

][][][][ nxnxnxnx ∗∗Associative

][][][][ 1221 nxnxnxnx ∗=∗

Di t ib ti

][][][][])[][(][ 3121321 nxnxnxnxnxnxnx ∗+∗=+∗Distributive

])[][(][][])[][( 321321 nxnxnxnxnxnx ∗∗=∗∗


Properties of LTI Systems

StabilityStabilityif and only if, sum of magnitude of Impulse Response, h[n] is finite

∞<= ∑∞

nhS |][| ∞<= ∑−∞=n

nhS |][|


Properties of LTI SystemsCausalityy

if and only if Impulse Response,h[n] = 0 for all n < 0

h 0,0][ <= nnh


Properties of LTI Systems


Linear Constant-Coefficient Difference Equations

Finite Impulse Response (FIR)p p ( )For causal, FIR systems

00,0][ ><= MandnfornhConvolution reduces to

0 0 ,0][ >< Mandnfornh

∑−

−=1

][][][M

knxkhny

Infinite Impulse Respose (IIR)

∑=0

][][][k

knxkhny

∑∞

−∞=

−=k

knxkhny ][][][



An important class of LTI systems of the formMN

The output is not uniquely specified for a given

[ ] [ ]∑∑==

−=−M

0kk

N

0kk knxbknya

p q y p ginput

The initial conditions are requiredLinearity, time invariance, and causality depend on the initial conditionsIf initial conditions are assumed to be zero system is linear, time invariant, and causal

ExampleExampleMoving AverageDifference eqn.

]3n[x]2n[x]1n[x]n[x]n[y −+−+−+=

[ ] [ ] 1ba where knxbknya30

∑∑q


[ ] [ ] 1ba where knxbknya kk0k

k0k

k ==−=− ∑∑==


nynx ][:output ][ :input

∑ ∑= =

−=−N

k

M

mmk mnxbknya

0 0][][

ExamplerAccumulato : ][][ kxny

n

= ∑

][]1[1

kxnyn

k

k

=− ∑−

−∞=

−∞=

]1[][][][][1

nynxkxnxnyn

k

k

−+=+= ∑−

−∞=

∞=


][]1[][ nxnyny =−− X[n] is the difference of y[n]

Block Diagram


Stability, Causality, and Time invariance of systems describe by LCCD

LCCD:

∑ ∑N M

bk

nynx

][][

][ :output ][ :input

∑ ∑= =

−=−k m

mk mnxbknya0 0

][][

Particular cases

∑ −=M

m mnxbny ][][ i. ∑=m 0

∑ −=N

nnyanx ][][ii


∑=n

n nnyanx0

][][ ii.

StabilityCase i:

May be stable if

∑∑ |][||||][|][MM

bb

When

∞<−<=−= ∑∑==

|][| |||][|][00 m

mm

m mnxbmnxbny

∞<∞< |b| and ,|][| mnx


StabilityCase ii:

N

∑=

−=N

nn nnyanx

0][][ ii.

Eventhough |an | < ∞, the system is unstableE.g.

For a >= 1 this system is unstable


Linearity and CausalitySuppose that for a given input x[n] we have pp g p [ ]found one particular output sequence yP[n] so that a LCCD equation is satisfied.Th th ti ith th i t i The the same equation with the same input is satisfied by any output of the form

[n]y[n]yy[n] +Where yH[n] is any solution to the LCCD

[n]y[n]y y[n] HP +=

equation with zero input x[n] = 0.Remark: yP[n] and yH[n] are referred to as the particular and homogeneous solutions particular and homogeneous solutions respectively.


Linearity and CausalityA LCCD equation does not provide a unique q p qspecification of output for a given inputAuxiliary information or conditions are required to specify uniquely the output for a given inputto specify uniquely the output for a given inputExample

Let auxiliary information be in the form of N sequential output values Thensequential output values. Then,Later values can be obtained by rearranging LCCD equation as a recursive relation running forward in nn.Prior values can be obtained by rearranging LCCD equation as a recursive relation running backward in n.n.


LCCD equations as recursive procedures


LTI and causal


Linearity and causality


Time invariance


Linear, Time invariant, and Causal system


DIGITAL SIGNAL ROCESSING LECTURE 3 - University of...

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