DSP_ch2

DSP_ch2 - DSP EEE 407/591 by Andreas Spanias, Ph.D. Chapter...

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Copyright ©Andreas Spanias 1 DSP EEE 407/591 by Andreas Spanias, Ph.D. Chapter 2 spanias@asu.edu http://www.eas.asu.edu/~spanias Copyright ©Andreas Spanias 2 Discrete-time Linear Systems – Digital Filters x(n) h(n) y(n) The output is produced by convolving the input with the impulse response ) ( * ) ( ) ( ) ( ) ( m x m h m n x m h n y m = = −∞ = This operation can also involve a finite-length impulse response(FIR) sequence = = L m m n x m h n y 0 ) ( ) ( ) ( An FIR filter is programmed using a multiply-accumulate instruction
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Copyright ©Andreas Spanias 3 Some Definitions •A digital filter is linear if it has the property of generalized superposition. •A digital filter is causal if it non anticipatory, i.e., the present output does not depend on future inputs. •All real-time systems are causal. • Non-causalities arise in image processing where the signal indexes are spatial instead of temporal. • Unless otherwise stated all systems in this course will be assumed causal Copyright ©Andreas Spanias 4 Some More Definitions L b . 1 z T or ;unit delay x(n) x(n-1) x(n) b L x(n) x(n) x(n-1) x(n)+x(n-1) ;signal scaling by a filter coefficient ;signal addition x(n) x(n-1)
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Copyright ©Andreas Spanias 5 IIR Digital Filter Structure = = = M i i L i i i n y a i n x b n y 1 0 ) ( ) ( ) ( feedback ( ) n y () n x T 1 a M a 0 b L b ... ... . . . . . . . . . . + + + + - - - 1 b 2 a . T T T T Copyright ©Andreas Spanias 6 FIR Digital Filter Structure = = L i i i n x b n y 0 ) ( ) ( ( ) n y n x T T 0 b L b ... . . . . . + + + + 1 b .
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Copyright ©Andreas Spanias 7 Two i/p-o/p Equations for Digital Filters x(n) h(n) y(n) One can compute the output using the convolution sum −∞ = −∞ = = = m m m n h m x m n x m h n y ) ( ) ( ) ( ) ( ) ( or by using the difference equation = = = M i i L i i i n y a i n x b n y 1 0 ) ( ) ( ) ( Remark: The impulse response h(n) can be determined by solving the difference equation. Copyright ©Andreas Spanias 8 Unit Impulse The analysis of digital filters in the frequency domain is facilitated using sinusoids. In the time domain a unique input signal is used for analysis, namely the unit impulse. That is defined as: () { 0 n for 1 elsewhere 0 = = n δ ( ) n n 0
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Copyright ©Andreas Spanias 9 Signal Representation with Unit Impulses Any discrete-time signal may be represented by a linear combination of unit impulses is represented by: xn n n n nn n () . ( ) . ( ) . ( ) . ( ) =− + + + + −− + 52 1 51 2 15 2 2 5 3 δ δδ ( ) n x n 1 0 2 3 4 5 5 0 . 2 5 1 . 5 0 . 5 2 . 1 Copyright ©Andreas Spanias 10 Impulse Response The response of a digital filter to a unit impulse is known as impulse response and is given by ) ( ... ) 2 ( ) 1 ( ) ( ... ) 1 ( ) ( ) ( 2 1 1 0 M n h a n h a n h a L n b n b n b n h M L + + + = δ (n) h(n) h(n)
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Copyright ©Andreas Spanias 11 Finite-Length Impulse Response (FIR) If the digital filter has no feedback terms the impulse response is finite length ) ( ... ) 1 ( ) ( ) ( 1 0 L n b n b n b n h L + + + = δ 0 ) 0 ( b h = 1 ) 1 ( b h = 2 ) 2 ( b h = L b L h = ) ( ..
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DSP_ch2 - DSP EEE 407/591 by Andreas Spanias, Ph.D. Chapter...

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