Part II. Linear Equalizations Matched-Filter, Zero-Forcing, MMSE - PowerPoint PPT Presentation

Part II. Linear Equalizations Matched-Filter, Zero-Forcing, MMSE Equalization 1

Mitigate ISI with Linear Filters { W m } Linear Equalizer symbol-wise { V m } { ˆ u m } LTI filter: { g ` } detection • ISI is caused by a (discrete-time) LTI filter due to the frequency selectivity of the channel • Why not use another discrete-time LTI filter at the receiver to mitigate ISI, and do symbol-wise detection at the filtered output? • Design of the filter requires some objectives for optimization: { g ` } ‣ Probability of error? hard to analyze ‣ Energy will be easier to handle • Since the ISI is treated as noise in the symbol-wise detection, we should try to maximize the signal-to-interference-and-noise ratio (SINR) at the filtered output { W m } 2

Linear Equalizers to be Introduced • Use Z transform to represent the discrete-time LTI filter X g ` ζ − ` g ( ζ ) , → ˇ g ` ← ` ‣ Recall its relation with DTFT: g ( e j2 π f ) g ( f ) = ˇ ˘ • Three kinds of linear equalizers: ‣ Matched filter (MF): g ( �� ) ( ζ ) = ˇ ˇ h ∗ (1 / ζ ∗ ) . ‣ Zero forcing (ZF): g ( �� ) ( ζ ) = (ˇ h ( ζ )) − 1 . ˇ ‣ Minimum mean squared error (MMSE): maximize SINR E s ˇ h ∗ (1 / ζ ∗ ) g ( ���� ) ( ζ ) = ˇ N 0 + E s ˇ h ∗ (1 / ζ ∗ )ˇ h ( ζ ) • Low SNR regime ( ): g ( ���� ) ( ζ ) ≈ E s g ( �� ) ( ζ ) ˇ N 0 ˇ E s ⌧ N 0 • High SNR regime ( ): g ( ���� ) ( ζ ) ≈ ˇ g ( �� ) ( ζ ) E s � N 0 ˇ 3

� � � � � � � � � Matrix Representation of ISI Channel V 1 h 0 u 1 Z 1 = + V 2 h 0 u 2 h 1 u 1 Z 2 = + + V L h 0 u L h 1 u L − 1 h L − 1 u 1 Z L = + + + + · · · V L +1 h 0 u L +1 h 1 u L h L − 1 u 2 Z L +1 = + + + + · · · V n h 0 u n h 1 u n − 1 h L − 1 u n − L +1 Z n = + + + + · · · V n +1 h 1 u n h L − 1 u n − L +2 Z n +1 = + + + · · · V n + L − 1 h L − 1 u n Z n + L − 1 = + 4

� � � � � � � � � � � � � � � � � � � � � Matrix Representation of ISI Channel � V = h u + Z = u m [ h ] m + u i [ h ] i + Z i ̸ = m  h 0  0 0 · · ·   h 1 h 0   m ∼ ( m + L − 1) -th     elements are h 0 , h 1 , ...h L − 1 h 1       h L − 1 0 h �     0 h L − 1 h 0       h 1 0         h L − 1 0 0 [ h ] m 5

Matrix Representation of Equalizer { W m } Linear Equalizer symbol-wise { V m } { ˆ u m } LTI filter: { g ` } detection W m = ⟨ V , [ g ] m ⟩ = [ g ] H m V ˜ Z m � [ g ] H m Z m [ h ] i ) u i + ˜ � = ([ g ] H ([ g ] H m [ h ] m ) u m + Z m i ̸ = m signal ISI noise | ⟨ [ h ] m , [ g ] m ⟩ | 2 E s Goal: maximize SINR = i ̸ = m | ⟨ [ h ] i , [ g ] m ⟩ | 2 E s + ∥ [ g ] m ∥ 2 N 0 � 6

Low SNR Regime E s ⌧ N 0 = ) ������� ���� m [ h ] i ) u i + ˜ � W m = ([ g ] H ([ g ] H m [ h ] m ) u m + Z m i ̸ = m | ⟨ [ h ] m , [ g ] m ⟩ | 2 E s SINR = i ̸ = m | ⟨ [ h ] i , [ g ] m ⟩ | 2 E s + ∥ [ g ] m ∥ 2 N 0 � � 2 E s � | ⟨ [ h ] m , [ g ] m ⟩ | = ∥ [ g ] m ∥ N 0 ⇒ [ g ( �� ) ] m = [ h ] m = 7

Matched Filter W m = h ∗ 0 V m + h ∗ 1 V m +1 + . . . + h ∗ L − 1 V m + L − 1 L − 1 0 0 g ( �� ) X X X h ∗ h ∗ ` V m + ` = − ` V m − ` = V m − ` , = ` ` =0 ` = − ( L − 1) ` = − ( L − 1) g ( �� ) ( ζ ) = ˇ ⇒ g ( �� ) ˇ h ∗ (1 / ζ ∗ ) = h ∗ = − ` ` g ( �� ) ( f ) = ˘ ˘ h ∗ ( f ) project the signal onto the signal direction, so that the signal energy is maximized. 8

High SNR Regime E s � N 0 = ) ������� ������ m [ h ] i ) u i + ˜ � W m = ([ g ] H ([ g ] H m [ h ] m ) u m + Z m i ̸ = m | ⟨ [ h ] m , [ g ] m ⟩ | 2 E s SINR = i ̸ = m | ⟨ [ h ] i , [ g ] m ⟩ | 2 E s + ∥ [ g ] m ∥ 2 N 0 � ⇒ [ g ( �� ) ] m ⊥ [ h ] i , ∀ i ̸ = m = [ g ( �� ) ] m = ( h † ) H e m = h ( h H h ) − 1 e m one choice: 9

Geometric Interpretation [ g ( �� ) ] m ≡ [ h ] m [ g ( �� ) ] m interference subspace �������� ������� �� ��� n − 1 ������� �� h ������� [ h ] m � 10

Max. SINR Min. MSE ≡ Linear Equalizer { V m } { W m } L − 1 X X X X W m = g k V m − k = g k h ` u m − k − ` + g k Z m − k k k ` =0 k �� L − 1 � u m + ˜ I m + ˜ = ℓ =0 g − ℓ h ℓ Z m the same for all m � WLOG assume it is 1 = u m + ˜ I m + ˜ Z m Ξ m : kind of estimation error ⇥ | U m | 2 ⇤ SINR = E E s h | Ξ m | 2 i max SINR ≡ min E E [ | Ξ m | 2 ] = E [ | Ξ m | 2 ] mean squared error (MSE) 11

Minimum MSE Estimation • In general, one can consider the following estimation problem: ‣ Given a random observation, estimate a target s.t. the MSE is minimized target observation estimation Estimator in H H P X,Y ˆ X = g ( Y ) X Y g ( · ) random processes LTI filter (FIR/IIR, causal/non-causal) { X n } , { Y n } random vectors general functions/linear functions X , Y � 2 � � MSE ( X, ˆ � X − ˆ g ( ���� ) ( · ) = argmin X ) , E MSE ( X, g ( Y )) X � � � g ∈ H ‣ You might be familiar with the general case: g ( ���� ) ( Y ) = E [ X | Y ] • Here, we focus on the random process case and linear estimators without any causality and finite-tap constraints. ‣ After deriving the optimal filter for MMSE estimation, we apply it back to the original problem 12

Recap: Discrete-Time Random Process General (joint) WSS First µ X [ n ] ≡ µ X µ X [ n ] , E [ X n ] moment Second R X [ n 1 , n 2 ] , E ⇥ ⇤ R X [ n + k, n ] ≡ R X [ k ] X n 1 X ∗ moment n 2 (auto-correlation) R X [ − k ] = R ∗ X [ k ] R XY [ n 1 , n 2 ] , E ⇥ ⇤ R XY [ n + k, n ] ≡ R XY [ k ] X n 1 Y ∗ n 2 (cross-correlation) R Y X [ k ] = R ∗ XY [ − k ] PSD R X [ k ] ← → S X ( ζ ) R XY [ k ] ← → S XY ( ζ ) S Y X ( ζ ) = S ∗ XY (1 / ζ ∗ ) 13

Recap: Filtering Random Processes X 1 [ n ] h 1 [ n ] Y 1 [ n ] = ( X 1 ∗ h 1 )[ n ] jointly WSS jointly WSS X 2 [ n ] h 2 [ n ] Y 2 [ n ] = ( X 2 ∗ h 2 )[ n ] Cross-correlation: R Y 1 ,Y 2 [ k ] = ( h 1 ∗ R X 1 ,X 2 ∗ h 2 , rv ) [ k ] S Y 1 ,Y 2 ( ζ ) = ˇ h 1 ( ζ ) S X 1 ,X 2 ( ζ )ˇ Cross PSD: h ∗ 2 (1 / ζ ∗ ) 14

Derivation of the Optimal Filter Estimation via jointly WSS Linear Filter { ˆ { Y n } { X n } X n } = { ( g ∗ Y ) n } { g k } ← → ˇ g ( ζ ) � 2 � � � X n − ˆ MSE , E X n � � � Ξ n { g ( ���� ) Goal: } = argmin MSE also WSS! k { g k } ! ∗ # " h X n ) ∗ i ( X n − ˆ X n )( X n − ˆ X Note: MSE = E = E Ξ n X n − g k Y n − k k ∂ ⇥ ⇤ ⇥ ⇤ ⇥ ⇤ ∀ k, 0 = MSE = − E Ξ n Y ∗ = E ( g ∗ Y ) n Y ∗ X n Y ∗ − E n − k n − k n − k ∂ g ∗ k ⇒ ˇ g ( ζ ) S Y ( ζ ) = S XY ( ζ ) ⇒ ∀ k, ( g ∗ R Y )[ k ] = R XY [ k ] ⇐ ⇐ Solution: g ( ���� ) ( ζ ) = ( S Y ( ζ )) − 1 S XY ( ζ ) (non-causal IIR Wiener filter) ˇ 15

Orthogonality Principle • A key equation in deriving the optimal estimator is ⇥ ⇤ Ξ n Y ∗ = 0 , 8 k ( ) h Ξ n , ( f ⇤ Y ) n i = 0 , 8 ��� ����� { f ` } E n − k ‣ For two r.v.’s , we define the “inner product” as h X, Y i , E [ XY ∗ ] ( X, Y ) ‣ (you can check the axioms of inner product space …) • A geometric interpretation: for an estimator that minimizes MSE, its estimation error should be “orthogonal” to the any estimators that one can choose ‣ Caveat: the family of estimators (which are also r.v.‘s) should form a “subspace” of the r.v. inner product space 16

target X Ξ ���� ˆ X ���� ( Y ) estimator subspace observation target estimation Estimator in H H P X,Y ˆ X = g ( Y ) X Y g ( · ) 17

The Minimum MSE min MSE = E [ Ξ n Ξ ∗ n ] = E [ Ξ n X ∗ n ] h i ( g ( ���� ) ∗ Y ) n X ∗ = E [ X n X ∗ n ] − E n g ( ���� ) X = R X [0] − R Y X [ − k ] k k = R X [0] − ( g ( ���� ) ∗ R Y X )[0] 1 Z 2 ⇣ ⌘ g ( ���� ) ( f ) S Y X ( f ) = S X ( f ) − ˘ d f − 1 2 ! 1 S X ( f ) − | S XY ( f ) | 2 Z 2 = d f S Y ( f ) − 1 2 18

Other kinds of Wiener Filter FIR Wiener Filter IIR Causal Wiener Filter 19

Optimal Linear Equalizer Back to our problem of linear equalization V m = ( h ∗ U ) m + Z m S U ( ζ ) = E s Linear Equalizer { U m } { V m } { W m } S Z ( ζ ) = N 0 { ˆ X n } { Y n } { X n } ⇒ S V ( ζ ) = ˇ h ( ζ ) S U ( ζ )ˇ V m = ( h ∗ U ) m + Z m = h ∗ (1 / ζ ∗ ) + S Z ( ζ ) S UV ( ζ ) = S U ( ζ )ˇ h ∗ (1 / ζ ∗ ) Optimal linear equalizer: E s ˇ g ( ���� ) ( ζ ) = S UV ( ζ ) h ∗ (1 / ζ ∗ ) ˇ S V ( ζ ) = E s ˇ h ∗ (1 / ζ ∗ )ˇ h ( ζ ) + N 0 20

The Maximum SINR 1 E s = max SINR = 1 ◆ − 2 min MSE 2 E s ✓� Z 2 � � ˘ h ( f ) + 1 d f � � N 0 � − 1 2 ! 1 S U ( f ) − | S UV ( f ) | 2 Z 2 min MSE = d f S V ( f ) − 1 2 2 0 1 � � � ˘ E 2 h ( f ) 1 � � Z 2 s � = A d f @ E s − B C 2 � � � ˘ − 1 h ( f ) E s + N 0 � � 2 � 2 0 1 1 Z 1 2 = E s d f B C 2 E s @ � � A � ˘ − 1 h ( f ) N 0 + 1 � � 2 � 21