Part III. OFDM Discrete Fourier Transform; Circular Convolution; - PowerPoint PPT Presentation

Part III. OFDM Discrete Fourier Transform; Circular Convolution; Eigen Decomposition of Circulant Matrices 1

Motivation { Z m } LTI Filter Channel { V m } { u m } { h ` } • Previous parts: only receiver-centric methods ‣ MLSD with Viterbi algorithm: optimal but computationally infeasible ‣ Linear equalizations: simple but suboptimal. • Is it possible to “pre-process” at Tx and “post-process” { u m } { V m } at Rx, so that the end-to-end channel is ISI-free? ‣ Note: ZF can already remove ISI completely, but the noises after ZF are not independent anymore ‣ Post processing should preserve mutual independence of the noises • Observation: IDTFT and DTFT will work, “if” we are willing to roll back to analog communication 2

Z 1 / 2 u ( f ) e j2 π mf d f IDTFT IDTFT: u m = ˘ u ( f ) ˘ { u m } − 1 / 2 { h ` } { Z m } ˘ ˘ X DTFT: V m e − j2 π mf DTFT V ( f ) = V ( f ) { V m } m → ˘ V ( f ) = ˘ u ( f ) + ˘ V m = ( h ∗ u ) m + Z m ← h ( f )˘ Z ( f ) In frequency domain, the outcome at a Why it works: because is e j2 π mf frequency only depends on the input at an eigenfunction to any LTI filter. that frequency: Using these eigenfunctions as a ⟹ no ISI! new basis to carry data renders infinite # of ISI-free channels in the frequency domain. Caveat: analog communication in the frequency domain 3

Discretized DTFT: Discrete Fourier Transform Idea: use the discretized version of DTFT/IDTFT Z 1 / 2 N − 1 1 u ( f ) e j2 π mf d f u [ k ] e j2 π mk X IDTFT: N -pt. IDFT: u m = ˘ u m = ˘ N f = k √ N − 1 / 2 k =0 N N − 1 1 k = 0 , ..., N − 1 V m e − j2 π mk ˘ ˘ X X DTFT: V m e − j2 π mf N -pt. DFT: V [ k ] = V ( f ) = N m = 0 , ..., N − 1 √ N m =0 m Note: N -point DFT/IDFT are transforms between two length- N sequences, indexed from 0 to N –1 . Unfortunately, the convolution-multiplication property of the DTFT-IDTFT pair no longer holds We need a new kind of convolution for DFT-IDFT pair! 4

Circular Convolution Definition: for two length- N sequences { x n } N − 1 n =0 , { h n } N − 1 n =0 N − 1 X ( h ~ x ) n , n = 0 , 1 , ..., N − 1 h ` x ( n − ` ) mod N , ` =0 h 0 h 0 n = 0 n = 2 h N − 1 h 1 h N − 1 h 1 x 0 x 2 x 1 x 1 x N − 1 h 2 h 2 x 0 x 2 x N − 1 Convolution-multiplication property: for length- N sequences { x n } N − 1 n =0 , { y n } N − 1 n =0 , { h n } N − 1 with , y n = ( h ~ x ) n n =0 √ N ˘ y [ k ] = ˘ h [ k ]˘ x [ k ] , ∀ k = 0 , 1 , ..., N − 1 5

Implement Circular Conv. in LTI Channel Original LTI channel (ignore noise) : linear convolution L − 1 X v m = ( h ∗ u ) m = h ` u m − ` , m = 0 , ..., N − 1 ( N � L ) ` =0 h 0 h 1 h L − 1 · · · u N − L u N − L u N − 1 u 0 u N − 1 · · · · · · · · · +1 +1 cyclic prefix Desired channel (ignore noise) : circular convolution L − 1 X v m = ( h ~ u ) m = h ` u ( m − ` ) mod N , m = 0 , ..., N − 1 ` =0 6

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � add cyclic prefix transmit receive remove cyclic prefix u N − L +1 x 1 y 1 h L − 1 CP u N − 1 x L − 1 y L − 1 h 0 u 0 u 0 x L y L v 0 v 0 x L +1 y L +1 u 1 u 1 v 1 v 1 x N + L − 1 y N + L − 1 u N − 1 u N − 1 v N − 1 v N − 1 convolution v m = ( h ~ u ) m , m = 0 , ..., N − 1 7

� � � � � � � � � � � � � � � � � � � � � � � � � � � Matrix Form of Circular Convolution v m = ( h ~ u ) m , m = 0 , ..., N − 1 ∈ C N v = h c u   h 0 h L − 1 h 1 0 0 v 0 u 0 · · ·   h 1 h 0   v 1 u 1 0 h L − 1  h 1    h 0 h L − 1 0 =     h 1 h 0 h L − 1 0     0   h L − 1 h L − 2 h 0 0 0 v N − 1 u N − 1 · · · h c v u with noise: ∈ C N V = h c u + Z 8

� � � � � � � � � � � � � � � � � � � � � Linear Algebraic View Circulant Matrix Every row/column is a circular shift of the first row/column   h 0 h L − 1 h 1 0 0 · · ·   h 1 h 0   0 h L − 1  h 1    h 0 h L − 1 0     h 1 h 0 h L − 1 0     0   h L − 1 h L − 2 h 0 0 0 · · · h c φ ( k ) N e j2 π k 1 Define m � N m , m = 0 , ..., N − 1 √ h [ k ] φ ( k ) N ˘ ( h ⊛ φ ( k ) ) m = √ Can show: for any , { h ℓ } N − 1 m ℓ =0 9

� � � � � � � � � � � � � � � � � � � � ���� ���������� � � � � √ h [ k ] φ ( k ) N ˘ ( h ⊛ φ ( k ) ) m = m = 0 , ..., N − 1 m , ⇒ φ ( k ) �� �� ����������� �� ������ h c = √ N ˘ h [ k ] � ��� ��� k = 0 , ..., N − 1 �     φ ( k ) h 0 h L − 1 h 1 0 0 · · · 0     h 1 h 0     φ ( k ) 0 h L − 1  h 1    1 N ˘     √ φ ( k ) h [ k ] h 0 h L − 1 0 =         h 1 h 0 h L − 1 0 ˘     h ( f ) | f = k     N 0     φ ( k ) h L − 1 h L − 2 h 0 0 0 · · · N − 1 φ ( k ) h c √ h c φ ( k ) = N ˘ h [ k ] φ ( k ) , ∀ k = 0 , ..., N − 1 Furthermore, can show that ⟨ φ ( k ) , φ ( l ) ⟩ = { k = l } ⇒ { φ ( k ) | k = 0 , ..., N − 1 } : an orthonormal basis of C N = 10

Hence, we can obtain the eigenvalue decomposition of any circulant matrix h c h Φ H h c = ΦΛ ˘ � φ (0) ... φ ( N − 1) � Φ � h � diag(˘ h ( f 0 ) , ˘ h ( f 1 ) , ..., ˘ h ( f N − 1 )) Λ ˘ f k � k N , k = 0 , ..., N − 1 { h n | n = 0 , ..., N − 1 } � ��� ���� ������ �� h c ˘ h ( f ) � ���� �� { h n } Can diagonalize the channel (remove ISI) without knowing it using the DFT basis. Only true for circulant matrix! 11

������ IDFT matrix and DFT matrix : Φ H Φ N − 1 1 u [ k ] e j2 π mk X N -pt. IDFT: u m = ˘ j2 π mk 1 � � N ( Φ ) m,k = N exp √ N √ N k =0 N − 1 − j2 π mk 1 � � ( Φ H ) m,k = N exp 1 V m e − j2 π mk ˘ X N -pt. DFT: V [ k ] = √ N N √ N m =0 ˘ V = Φ H V u = Φ ˘ u , Pre-processing and post-processing: h Φ H u + Z Φ H V = Λ ˘ h Φ H u + Φ H Z V = h c u + Z = ΦΛ ˘ ⇒ ˘ u + ˘ ˘ V = Λ ˘ h ˘ ∼ CN (0 , N 0 ) , k = 0 , 1 , ..., N − 1 = Z Z [ k ] because the DFT matrix is unitary Φ H 12

Equivalent Parallel Channels OFDM creates N parallel non-interfering sub-channels: V [ k ] = ˘ ˘ u [ k ] + ˘ h ( f k )˘ Z [ k ] , k = 0 , 1 , ..., N − 1 Channel gain at the k -th branch: h ( f k ) = ˘ ˘ N ˘ h ( k √ ˘ h ( f ) : DTFT of { h ` } N ) = h [ k ] = periodic copies of ˘ h a ( f T ) , period 1 h a ( τ ) � ( h b ∗ g )( τ ) Equivalently, the overall bandwidth 2 W is partitioned into N narrowbands, and each sub-channel use that narrowband for transmission (centered at ) k 2 W N , k = 0 , ..., N − 1 Subcarrier spacing: 2 W N 13

Capacity of Parallel Channels Capacity of N parallel channels is ˘ ˘ h ( f 0 ) Z [0] the sum of individual capacities coding across subcarriers does not help! ˘ u [0] ˘ V [0] Since channel gains are di ff erent, ˘ ˘ h ( f 1 ) Z [1] each branch has di ff erent capacity ˘ u [1] ˘ V [1] Goal: maximize capacity subject . . to a total power constraint . . . . Power allocation: maximize rate N − 1 P k : power of branch k X ˘ P k ≤ NP ˘ h ( f N − 1 ) Z [ N − 1] k =0 N − 1 � � 2 P k 1 + | ˘ h ( f k ) | ˘ u [ N − 1] ˘ � V [ N − 1] R = log N 0 k =0 14

������� �� Water-filling N − 1 2 P k ✓ ◆ � � � ˘ X max log 1 + h ( f k ) , � � N 0 � P 0 ,...,P N − 1 k =0 N − 1 X P k = NP, P k ≥ 0 , k = 0 , . . . , N − 1 n =0 Solved by standard techniques in convex optimization (Lagrange multipliers, KKT condition) Final solution: � + � N 0 P ∗ ν − k = ( x ) + � max(0 , x ) 2 | ˘ h ( f k ) | � + N − 1 � N 0 � = NP ν �������� ν − 2 | ˘ h ( f k ) | k =0 15

Total Area = P ν P ∗ P ∗ L − 1 P ∗ k 0 · · · N 0 | ˘ h ( f k ) | 2 k Main lesson: one should h ( f k ) = ˘ ˘ h b ( k 2 W g ( k 2 W N )˘ N ) allocate higher rate when baseband frequency response at the branch with better at f = k 2 W channel condition N 16

������� �� Capacity of Frequency Selective Channel Pre-processing (IDFT) and post-processing (DFT) are both invertible in OFDM systems The only loss: length- ( L –1) cyclic prefix, negligible when we take N → ∞ The power allocation problem becomes Z 1 / 2 2 P ( f ) ✓ ◆ � � � ˘ max log 1 + h ( f ) d f, � � N 0 � P ( f ) − 1 / 2 Z 1 / 2 P ( f ) d f = P, P ( f ) ≥ 0 , f ∈ [ − 1 / 2 , 1 / 2] − 1 / 2 Optimal solution: water-filling on the continuous spectrum 17

Water-filling in Frequency-Selective Channel Total Area = P ν N 0 | ˘ h a ( f ) | 2 k + W − W 18