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DSP: All-Pass Systems Digital Signal Processing All-Pass Systems D. Richard Brown III D. Richard Brown III 1 / 7

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DSP: All-Pass Systems All-Pass System Basics Deﬁnition A stable LTI system with rational transfer function is called all-pass if its DTFT satisﬁes j for some value A > and all All-pass systems only aﬀect the phase of the signal. They are of ten used to compensate for undesired phase shifts or delays in other sy stems without aﬀecting the shape of the magnitude response. Example: The delay system ) = is clearly an all-pass system since j ) = jωn and j = 1 Remarks: All-pass systems typically do not have linear phase. Some textbooks deﬁne all-pass systems such that j = 1 D. Richard Brown III 2 / 7

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DSP: All-Pass Systems All-pass Delay Equalizer Example It is not possible to design a stable causal IIR ﬁlter with lin ear phase. So what we can do instead is cascade an all-pass ﬁlter with an IIR ﬁlter to get approximately linear phase over a desired range of frequencies. Example: 0.5 1.5 2.5 3.5 0.5 1.5 magnitude response LPF APF cascade LPF−>APF 0.5 1.5 2.5 3.5 −15 −10 −5 phase response (rad) 0.5 1.5 2.5 3.5 normalized freq (rad/sample) group delay (samples) D. Richard Brown III 3 / 7

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DSP: All-Pass Systems Properties of All-Pass Systems Since we require j for some value A > and all , it is equivalent to write j j j ) = ] = (1 /z ) = Interpretation: must have the property that (1 /z is its inverse system (to a scale factor). This means that is all-pass if and only if the poles of are canceled by the zeros of (1 /z and the zeros of are canceled by the poles of (1 /z D. Richard Brown III 4 / 7

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DSP: All-Pass Systems First-Order All-Pass System As an example, suppose ) = 1 az Then (1 /z ) = (1 (1 /z = 1 z. Hence, we can form a ﬁrst-order all-pass ﬁlter as ) = (1 /z az Since (1 /z ) = (1 /z it should be clear that (1 /z ) = 1 Remarks: We have poles at and = 0 We have zeros at and In general, all-pass ﬁlters have conjugate-reciprocal pole-ze ro pairs. D. Richard Brown III 5 / 7

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DSP: All-Pass Systems All-pass Filter Poles and Zeros Mirrored Across Unit Circle −2 −1.5 −1 −0.5 0.5 1.5 2.5 −2 −1.5 −1 −0.5 0.5 1.5 Real Part Imaginary Part D. Richard Brown III 6 / 7

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DSP: All-Pass Systems General All-Pass Systems Given a ﬁrst-order all-pass ﬁlter ) = (1 /z az note that cz is also all-pass if c > Also observe that the cascade of two or more all-pass ﬁlters is al l-pass. Hence, transfer functions of the form ) = cz where each pole is paired with a conjugate reciprocal zero are all-p ass. The phase response follows as j ) = jωn (1 j j j See Matlab functions isallpass and iirgrpdelay D. Richard Brown III 7 / 7

Richard Brown III D Richard Brown III 1 7 brPage 2br DSP AllPass Systems AllPass System Basics De64257nition A stable LTI system with rational transfer function is called allpass if its DTFT satis64257es j for some value A and all Allpass systems ID: 22140

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Page 1

DSP: All-Pass Systems Digital Signal Processing All-Pass Systems D. Richard Brown III D. Richard Brown III 1 / 7

Page 2

DSP: All-Pass Systems All-Pass System Basics Deﬁnition A stable LTI system with rational transfer function is called all-pass if its DTFT satisﬁes j for some value A > and all All-pass systems only aﬀect the phase of the signal. They are of ten used to compensate for undesired phase shifts or delays in other sy stems without aﬀecting the shape of the magnitude response. Example: The delay system ) = is clearly an all-pass system since j ) = jωn and j = 1 Remarks: All-pass systems typically do not have linear phase. Some textbooks deﬁne all-pass systems such that j = 1 D. Richard Brown III 2 / 7

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DSP: All-Pass Systems All-pass Delay Equalizer Example It is not possible to design a stable causal IIR ﬁlter with lin ear phase. So what we can do instead is cascade an all-pass ﬁlter with an IIR ﬁlter to get approximately linear phase over a desired range of frequencies. Example: 0.5 1.5 2.5 3.5 0.5 1.5 magnitude response LPF APF cascade LPF−>APF 0.5 1.5 2.5 3.5 −15 −10 −5 phase response (rad) 0.5 1.5 2.5 3.5 normalized freq (rad/sample) group delay (samples) D. Richard Brown III 3 / 7

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DSP: All-Pass Systems Properties of All-Pass Systems Since we require j for some value A > and all , it is equivalent to write j j j ) = ] = (1 /z ) = Interpretation: must have the property that (1 /z is its inverse system (to a scale factor). This means that is all-pass if and only if the poles of are canceled by the zeros of (1 /z and the zeros of are canceled by the poles of (1 /z D. Richard Brown III 4 / 7

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DSP: All-Pass Systems First-Order All-Pass System As an example, suppose ) = 1 az Then (1 /z ) = (1 (1 /z = 1 z. Hence, we can form a ﬁrst-order all-pass ﬁlter as ) = (1 /z az Since (1 /z ) = (1 /z it should be clear that (1 /z ) = 1 Remarks: We have poles at and = 0 We have zeros at and In general, all-pass ﬁlters have conjugate-reciprocal pole-ze ro pairs. D. Richard Brown III 5 / 7

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DSP: All-Pass Systems All-pass Filter Poles and Zeros Mirrored Across Unit Circle −2 −1.5 −1 −0.5 0.5 1.5 2.5 −2 −1.5 −1 −0.5 0.5 1.5 Real Part Imaginary Part D. Richard Brown III 6 / 7

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DSP: All-Pass Systems General All-Pass Systems Given a ﬁrst-order all-pass ﬁlter ) = (1 /z az note that cz is also all-pass if c > Also observe that the cascade of two or more all-pass ﬁlters is al l-pass. Hence, transfer functions of the form ) = cz where each pole is paired with a conjugate reciprocal zero are all-p ass. The phase response follows as j ) = jωn (1 j j j See Matlab functions isallpass and iirgrpdelay D. Richard Brown III 7 / 7

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