Optimum IIR Filters Introduction and Scope • Definitions • Discussed FIR filters for both stationary and nonstationary cases • Design and properties • For simplicity and due to time constraints, will limit discussion of IIR filters to stationary case • Stationary case • IIR filters are not very useful practically, but can gain some • Frequency domain interpretations insights from them • Example • Does it make sense to use an IIR filter with a nonstationary signal? • One-step forward prediction = whitening J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 1 J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 2 Weiner-Hopf Equations Weiner-Hopf Equations MMSE Assuming h ( n ) is stable, Similarly, we can obtain the MMSE as follows ∞ ∞ � y ( n ) = ˆ h ( k ) x ( n − k ) � y ( n ) = ˆ h ( k ) x ( n − k ) k = −∞ k = −∞ 0 = E[( y ( n ) − ˆ y o ( n )) x ∗ ( n − ℓ )] by orthogonality P e o = E[( y ( n ) − ˆ y o ( n )) ( y ( n ) − ˆ y o ( n ))] ∞ = E[( y ( n ) − ˆ y o ( n )) y ∗ ( n )] � = E[ y ( n ) x ∗ ( n − ℓ )] − h o ( k ) E[ x ( n − k ) x ∗ ( n − ℓ )] ∞ k = −∞ � = E[ y ( n ) y ∗ ( n )] − h o ( k ) E [ x ( n − k ) y ∗ ( n )] ∞ � k = −∞ r yx ( ℓ ) = h o ( k ) r x ( ℓ − k ) ∞ k = −∞ � = r y (0) − h o ( k ) r xy ( − k ) • Last equation is the Weiner-Hopf equation k = −∞ • If causal or FIR, only applies for a finite range of ℓ ∞ � = r y (0) − h o ( k ) r ∗ yx ( k ) • Note use of projection theorem k = −∞ • This is a more general form of the normal equations, Rc o = d J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 3 J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 4
Weiner-Hopf Equations Impulse Response MSE If noncasual IIR, the Weiner-Hopf equations apply for all values of ℓ Alternatively, we can minimize the MSE. Given that and can be expressed as a convolution ˆ y ( n ) = h ( n ) ∗ x ( n ) ∞ � h o ( k ) r x ( ℓ − k ) = r yx ( ℓ ) h o ( ℓ ) ∗ r x ( ℓ ) = r yx ( ℓ ) where h ( n ) is not necessarily the optimal impulse response, find an expression for the MSE. Do not use the assumption that h ( n ) is FIR or k = −∞ causal in your derivation. H o = R yx (e jω ) H o (e jω ) R x (e jω ) = R yx (e jω ) R x (e jω ) � y ( n ) | 2 � • Thus we immediately obtain an expression for the optimal P e = E | y ( n ) − ˆ noncausal transfer function y ( n )) ∗ ( y ( n ) − ˆ � � = E ( y ( n ) − ˆ y ( n )) • It is more difficult to obtain the optimum impulse response, = E [( y ∗ ( n ) − ˆ y ∗ ( n )) ( y ( n ) − ˆ y ( n ))] though it is completely defined by the Weiner-Hopf equations = E [ y ∗ ( n ) y ( n )] − E [ y ∗ ( n )ˆ y ( n )] − E [ˆ y ∗ ( n ) y ( n )] + E [ˆ y ∗ ( n )ˆ y ( n )] ① ② ③ ④ J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 5 J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 6 Impulse Response MSE Continued Impulse Response MSE Continued P e = E [ y ∗ ( n ) y ( n )] − E [ y ∗ ( n )ˆ y ( n )] − E [ˆ y ∗ ( n ) y ( n )] + E [ˆ y ∗ ( n )ˆ y ( n )] = E [ y ∗ ( n )ˆ y ( n )] ② ① ② ③ ④ � � �� ∞ � = E y ∗ ( n ) h ( ℓ ) x ( n − ℓ ) � π E [ y ∗ ( n ) y ( n )] = r y (0) = 1 R y (e jω ) d ω ℓ = −∞ ① = 2 π ∞ − π � = h ( ℓ ) E [ y ∗ ( n ) x ( n − ℓ )] ℓ = −∞ ∞ � = h ( ℓ ) r ∗ yx ( ℓ ) ℓ = −∞ � ∞ ∞ 1 � H (e jω ) R ∗ yx (e jω ) d ω h ( ℓ ) r ∗ yx ( ℓ ) = 2 π −∞ ℓ = −∞ � ∞ ② ∗ = 1 H ∗ (e jω ) R yx (e jω ) d ω ③ = 2 π −∞ J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 7 J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 8
Impulse Response MSE Continued Impulse Response MSE Continued P e = E [ y ∗ ( n ) y ( n )] − E [ y ∗ ( n )ˆ y ( n )] − E [ˆ y ∗ ( n ) y ( n )] + E [ˆ y ∗ ( n )ˆ y ( n )] ∞ ∞ ① ② ③ ④ � � = h ∗ ( k ) h ( ℓ ) r x ( k − ℓ ) ④ k = −∞ ℓ = −∞ ④ = E [ˆ y ∗ ( n )ˆ y ( n )] ∞ � �� � � �� = h ∗ ( k ) ( h ( k ) ∗ r x ( k )) ∞ ∞ � � = E h ∗ ( k ) x ∗ ( n − k ) h ( ℓ ) x ( n − ℓ ) k = −∞ k = −∞ ℓ = −∞ � z ( k ) h ( k ) ∗ r x ( k ) � � ∞ ∞ � � ∞ = E h ∗ ( k ) h ( ℓ ) x ∗ ( n − k ) x ( n − ℓ ) � = h ∗ ( k ) z ( k ) ④ k = −∞ ℓ = −∞ k = −∞ ∞ ∞ � ∞ 1 � � = h ∗ ( k ) h ( ℓ ) E [ x ( n − ℓ ) x ∗ ( n − k )] H ∗ (e jω ) Z (e jω ) d ω = 2 π k = −∞ ℓ = −∞ −∞ � ∞ 1 ∞ ∞ H ∗ (e jω ) H (e jω ) R x (e jω ) d ω = � � = h ∗ ( k ) h ( ℓ ) r x ( k − ℓ ) 2 π −∞ k = −∞ ℓ = −∞ J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 9 J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 10 Impulse Response MSE Continued Re-examining the Stationary MSE Recall that in the FIR case we can express the MSE in terms of the ∞ ∞ cross-correlation matrix R and vector d � � P e = r y (0) − h ( ℓ ) r ∗ yx ( − ℓ ) − h ∗ ( ℓ ) r yx ( − ℓ ) | e ( n ) | 2 � � ℓ = −∞ ℓ = −∞ P e = E ∞ ∞ �� � H � �� y ( n ) − c H x ( n ) y ( n ) − c H x ( n ) � � = E + h ∗ ( k ) h ( ℓ ) r x ( k − ℓ ) k = −∞ ℓ = −∞ �� y ∗ ( n ) − x H ( n ) c � � y ( n ) − c H x ( n ) �� = E � ∞ 1 R y (e jω ) − H (e jω ) R ∗ yx (e jω ) − H ∗ (e jω ) R yx (e jω ) = � y ∗ ( n ) y ( n ) − x H ( n ) c y ( n ) − y ∗ ( n ) c H x ( n ) = E 2 π −∞ c H x ( n ) x H ( n ) c � � � �� + + H ∗ (e jω ) H (e jω ) R x (e jω ) d ω c − c H E [ x ( n ) y ∗ ( n )] | y ( n ) | 2 � x H ( n ) y ( n ) � � � = E − E + c H E x ( n ) x H ( n ) � � c • Recall we did not assume h ( n ) is FIR or causal, though it does P y − d H c − c H d + c H Rc = work in the case where c ∗ = h • This gives us two additional ways of calculating the MSE in terms of 1) the impulse response and 2) the frequency domain J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 11 J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 12
Frequency Domain MSE: Completing the Square Optimum IIR Stationary Filter Find the optimum IIR noncausal H (e jω ) by minimizing the MSE. � ∞ 1 R y (e jω ) + R yx (e jω ) R − 1 x (e jω ) R ∗ yx (e jω ) P e = � ∞ 1 2 π R y (e jω ) − H (e jω ) R ∗ yx (e jω ) − R yx (e jω ) H ∗ (e jω ) P e = −∞ � ∗ d ω 2 π R x (e jω ) H (e jω ) − R yx (e jω ) R − 1 x (e jω ) R x (e jω ) H (e jω ) − R yx (e jω ) � � � + −∞ + H (e jω ) R x (e jω ) H ∗ (e jω ) d ω • This is very similar to the approach we used to find the optimum � ∞ 1 R y (e jω ) + H (e jω ) R x (e jω ) H ∗ (e jω ) − R ∗ yx (e jω ) FIR filter in terms of R and d � � = 2 π −∞ • The MMSE is obtained by setting the last term to zero, − R yx (e jω ) H ∗ (e jω ) d ω H o (e jω ) = R yx (e jω ) � ∞ 1 R x (e jω ) as before R y (e jω ) + R yx (e jω ) R − 1 x (e jω ) R ∗ yx (e jω ) = 2 π −∞ � ∞ P e o = 1 H (e jω ) − R yx (e jω ) R − 1 x (e jω ) R x (e jω ) H ∗ (e jω ) − R ∗ yx (e jω ) R y (e jω ) + H o (e jω ) R ∗ yx (e jω ) d ω � � � � + d ω 2 π � ∞ −∞ 1 R y (e jω ) + R yx (e jω ) R − 1 x (e jω ) R ∗ yx (e jω ) = • Requires that R x (e jω ) be positive everywhere (no zeros on the 2 π −∞ unit circle) � ∗ d ω R − 1 � R x (e jω ) H (e jω ) − R yx (e jω ) � x (e jω ) � R x (e jω ) H (e jω ) − R yx (e jω ) + • Equivalently, r x ( ℓ ) must be positive definite (strictly) J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 13 J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 14 Frequency-Domain Minimum MSE The Role of Coherence � π P o = 1 If an optimum filter is used, 1 − G 2 � � R y (e jω ) d ω yx ( ω ) 2 π H o (e jω ) = R yx (e jω ) − π • Recall that coherence is a measure of correlation in the frequency R x (e jω ) domain then the MMSE is given by • If coherence is 1 at a frequency, the error at that frequency is zero � π P e o = 1 R y (e jω ) − H o (e jω ) R ∗ yx (e jω ) d ω • Similarly, if incoherent, the error power is equal to the output 2 π − π signal power � π = 1 1 − G 2 R y (e jω ) d ω � � yx ( ω ) • Applies to both causal IIR and FIR filters 2 π − π where G 2 yx ( ω ) is the magnitude squared coherence, | R xy (e jω ) | 2 G 2 yx ( ω ) � R x (e jω ) R y (e jω ) J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 15 J. McNames Portland State University ECE 539/639 Optimum FIR Filters Ver. 1.02 16
Recommend
More recommend
