F undam entals of D S P F undam entals of D S P L inear T im e ...

Linear Time-invariant Systems

Y H Chan

Fundamentals of DSP

Linear time-invariant systems
Transformation
Digital filters

CYH05-DAP-DSP

Linear time-invariant discrete (LTD) systems

CYH05-DAP-DSP




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CYH05-DAP-DSP

CYH05-DAP-DSP

CYH05-DAP-DSP

Definition:

Transformation

or, equivalently

The output y(n) is given by the convolution of the impulse response of the LTD system (h(n)) and the input (x(n)). Notation:


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CYH05-DAP-DSP

Z-transform

CYH05-DAP-DSP

• Z transform:

• Discrete Fourier transform:

• Fourier transform:

Transformation

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CYH05-DAP-DSP

CYH05-DAP-DSP

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CYH05-DAP-DSP

CYH05-DAP-DSP

CYH05-DAP-DSP

Digital Filters

(1)

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All bi=0

–

Not all bi=0

Infinite impulse response (IIR) filter

–

Finite impulse response (FIR) filter

CYH05-DAP-DSP







CYH05-DAP-DSP

• Transfer function:

• Z-transform of eqn.(1):

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(2)

Emphasize particular frequency component Attenuate particular frequency components

CYH05-DAP-DSP

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Filter design: Adjust poles and zeros of the filter to determine the frequency response of the filter.

–

Roots of polynomial X(z) are called poles.

–

Roots of polynomial Y(z) are called zeros.

CYH05-DAP-DSP










CYH05-DAP-DSP

Example: FIR

CYH05-DAP-DSP

Implementation of an FIR

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

(1z ) z =e jw = 14 (2 + e jw + e− jw ) = 12 (1 + cos w)

 H ( z )H



(1z ) =  z2+z11/2z/ +z 1 = 14 (2 + z + 1z ) H ( z )H

H e jw

( )2 ≡ H (z )H (1z ) z =e jw

Frequency response

CYH05-DAP-DSP

⇒




CYH05-DAP-DSP

Implementation of an IIR

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H ( z )H

CYH05-DAP-DSP

(

)

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17 + 4(e

j 2w

+e

)

− j 2w

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2w) ( ) z=e jw = 32 − 16(ej 2w + e− j 2w ) = 3217(1+−8cos cos 2w

(1z ) = 

2 − z 2 + z −2 z 2 − 1  z −2 − 1  = 2 −2 2  1  −2  z + 0.25  z + 0.25  16 (17 + 4( z + z ))

H e jw

( )2 ≡ H (z )H (1z ) z =e jw

Frequency response

⇒ H ( z )H 1 z




CYH05-DAP-DSP

CYH05-DAP-DSP

CYH05-DAP-DSP

Example: IIR

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