W Alexander C Williams NCSU FUNDAMENTAL DSP CONCEPTS ECE 513 Fall

W alexander c williams ncsu fundamental dsp concepts

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W. Alexander & C. Williams (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2019 62 / 170
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Difference Equation Representation The system transfer function, H ( z ) , can also be written in terms of the ratio of the Z Transforms of the output and the input. H ( z ) = Y ( z ) X ( z ) (67) It follows that Y ( z ) " 1 . 0 + L X k = 1 a ( k ) z - k # = X ( z ) " L X k = 0 b ( k ) z - k # (68) W. Alexander & C. Williams (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2019 63 / 170
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Difference Equation Representation The time shift property of the Z Transform can be used to obtain the difference equation representation for the discrete time system. If x ( n ) X ( z ) (69) then x ( n - k ) z - k X ( z ) (70) The region of convergence is the same as that for X ( z ) except at z = 0 for k > 0 and except for z = for k < 0 . Thus, if x ( n - k ) = 0 n < k , then z - k X ( z ) = X n = 0 x ( n - k ) z - n (71) W. Alexander & C. Williams (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2019 64 / 170
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Difference Equation Representation Similarly, if y ( n - k ) = 0 n < k , then z - k Y ( z ) = X n = 0 y ( n - k ) z - n (72) This result can be used to modify Equation 68 to obtain X n = 0 [ y ( n ) + a ( 1 ) y ( n - 1 ) + · · · + a ( L ) y ( n - L )] z - n (73) = X n = 0 [ b ( 0 ) x ( n ) + b ( 1 ) x ( n - 1 ) + · · · + b ( L ) x ( n - L )] z - n W. Alexander & C. Williams (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2019 65 / 170
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Difference Equation Representation The powers of z can be equated to obtain y ( n ) + a ( 1 ) y ( n - 1 ) + · · · + a ( L ) y ( n - L ) = b ( 0 ) x ( n ) + b ( 1 ) x ( n - 1 ) + · · · + b ( L ) x ( n - L ) (74) or y ( n ) = b ( 0 ) x ( n ) + b ( 1 ) x ( n - 1 ) + · · · + b ( L ) x ( n - L ) - a ( 1 ) y ( n - 1 ) - · · · - a ( L ) y ( n - L ) (75) W. Alexander & C. Williams (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2019 66 / 170
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Difference Equation Representation It follows that the linear shift invariant discrete time system can be represented in either of the following standard forms. H ( z ) = L X k = 0 b ( k ) z - k 1 . 0 + L X k = 1 a ( k ) z - k y ( n ) = L X k = 0 b ( k ) x ( n - k ) - L X k = 1 a ( k ) y ( n - k ) (76) Note the relationship between the coefficients b ( k ) and a ( k ) in the system transfer function and the corresponding coefficients in the difference equation. This relationship makes it easy to write the difference equation for a given system transfer function by inspection or vice versa. W. Alexander & C. Williams (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2019 67 / 170
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Frequency Response The general form for the transfer function for a causal, discrete time is H ( z ) = Y ( z ) X ( z ) = L X k = 0 b ( k ) z - k 1 . 0 + L X k = 1 a ( k ) z - k (77) Note that H ( z ) is represented using negative powers of z. W. Alexander & C. Williams (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2019 68 / 170
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Frequency Response H ( z ) can also be written in the form H ( z ) = Y ( z ) X ( z ) = L X k = 0 b ( k ) z L - k z L + L X k = 1 a ( k ) z L - k (78) This form can be obtained by multiplying the numerator and denominator of the previous form by z L . W. Alexander & C. Williams (NCSU) FUNDAMENTAL DSP CONCEPTS ECE 513, Fall 2019 69 / 170
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Frequency Response I The frequency response of a discrete time system can be obtained by applying a unit magnitude, sinusoidal sequence with zero phase and arbitrary frequency as an its input. A unit magnitude sinusoidal sequence with zero phase and arbitrary frequency can be represented as follows.
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