Complex Weighting Scheme for OFDM Cognitive Radio Systems

Transcription

1 Complex Weighting Scheme for OFDM Cognitive Radio Systems Yuhan Zhang Alireza Seyedi University of Rochester April 12, 2010

2 Outline 1 Introduction Background Using OFDM in Cognitive Radio Motivation and Problem Discerption 2 Optimized Complex Weighting System Model System Block Diagram System Model Description Partial Weighting Parameter Design Quadratic Approach BER Performance Simulation Result 3 Sequence Selection System Model Description Simulation Result 4 Conclusion 5 Future Work

3 Background Cognitive Radio and OFDM Cognitive Radios (CR) solve the spectral scarcity problem increase the overall spectrum usage by enabling secondary users Two Approaches of CR Overlay Design Underlay Design Orthogonal Frequency Division Multiplexing (OFDM) very flexible and agile in the frequency domain, and thus is a perfect candidate for the Underlay design of cognitive radios.

4 Using OFDM in Cognitive Radio OFDM is a Good Approach of Upperlay Design of CR. Advantage subcarriers that overlap with the bandwidth of the detected primary can be easily turned off limit the interference caused to the primary Disadvantage The sidelobes of subcarriers not overlapping the bandwidth can still cause a considerable level of interference to the primary user.

5 Using OFDM in Cognitive Radio OFDM is a Good Approach of Underlay Design of CR. Cont. Fig. 2 depicts the spectrum of an OFDM system with N = 64 subcarriers, where N notch = 10 subcarriers are turned off. The deepest point of the created notch is about 15dB, which is caused by the interference from the secondary user. 5 BPSK Signal with a Notch in the Middle Notch Depth/dB Index of OFDM Subcarriers

6 Motivation and Problem Discerption Methods Used to Cancel Interference Methods List Active Interference Cancelation (AIC) Windowing Real Subcarrier Weighting (RSW) Idea of RSW Weight the data in subcarriers which not overlap with the primary user. All the weights are real scalar. Optimize the weights that the sidelobes of different subcarriers cancel each other to the largest extent. Consequently, the interference to the primary user is reduced.

7 Motivation and Problem Discerption Problem Discerption Idea of Complex Weighting Extend the idea of RSW to allow complex subcarrier weights. Extend the all-subcarrier weighting to partial weighting. Consider the notch depth together with the BER performance to choose the best weights. The particular weights effect whether the sidelobes of different subcarriers add in a constructive or destructive manner. Using different sets of weights changes the notch depth. These weights are limited to be close to 1, so that detection can be done at the receiver without the knowledge of the weights themselves.

8 System Model How Does The System Work? The system block diagram can be described as Fig. 1. The weighting scheme is done between the S/P and IDFT.

9 System Model How to describe the system? Let us consider an OFDM system with N subcarriers. Assume: L start : The index of subcarrier where the overlapped bandwidth strats from. L end : The index of subcarrier where the overlapped bandwidth ends. N notch = L end L start + 1: Total number of subcarriers need to be turned off. Moreover, for brevity of notation, we define A on and A off to be sets of all indices of on and off sub-carriers, respectively.

10 System Model How to describe the system? Cont. As can be seen in Fig.3, the middle point of each sidelobe approximately contributes the most to the interference in the notched region.in order to minimize the interference, we would like to decrease these middle values Power Spectrum of the Signal/dB Index of ODFM Subcarrier Of course, the proposed method can be easily extended so that it attempts to reduce the interference at more points, e.g. at a quarter and three quarters of sub-carrier spacing. The calculation will be shown on next slide.

11 System Model Mathematic Calculation The value of the middle point of sidelobe between subcarriers m and m + 1 caused by the data in the nth sub-carrier: S nm+ 1 2 = d nw ni((m ) n), where d n is the nth data symbol, w n is the weight of the nth subcarrier, and I(x) = sin(πx) 1 N sin( πx ) e j(1 N )πx. N Thus the total value of interference at this point resulting from all on subcarriers can be calculated as Math Form Vector Form y(m ) = = n A on,m A on S nm+ 1 2 d nw ni((m + 1 ) n). (1) 2 n A on,m A on y = ADw = Bw, (2)

12 System Model Mathematic Calculation Cont. In Vector Form, we have y = [y 1... y Nnotch ] T, with y i = y(i + L start ), w = [w 1... w Lstart 1 w Lend w N ] T, D = diag([d 1... d Lstart 1 d Lend d N ]). A is the matrix of I functions, A = I(E F). E = [ e 1 e 2... e Nnotch ] T [ ], with e i = i + L start , i = 1,..., N notch. F = [ ] T [ f1 f 2... f N NNotch ], with f i = 1, 2,..., L start 1, L end + 1,..., N. A, D, y and w have complex components. An equivalent real description of the above can be written as y. It can be easily turned into all-real matrix by separating y i into real and image part.

13 System Model Mathematic Calculation Cont. y R = [ Re(y) Im(y) = B R w R, ] [ Re(B) Im(B) = Im(B) Re(B) ] [ Re(w) Im(w) ] (3) The total interference power in the notch is then given by minimize y R 2 = y T R y R = (B Rw R ) T (B R w R ) = w T RB T RB R w R.

14 Partial Weighting Why Partial Weighting? From Fig.3, it can be easily found that the sidelobes from subcarriers closer to the notch part have greater influence to the notch than the others. Instead of weighting all on subcarriers, just weight the subcarriers close to the notch can decrease the scare of calculation, while maintain a good notch depth. The power spectrum of the overlapped subcarriers now can be divided into two parts. Interference from the unweighted subcarriers Interference from those weighted subcarriers We define C uw and C w to be sets of all indices of weighted and unweighted subcarriers, respectively, C uw + C w = C on.

15 Partial Weighting Mathematic Calculation Equation 1 can be rewritten in vector form as y = A uwd uww uw + A wd ww w = B uww uw + B ww w, (4) where w uw = [w uw1... w uwi ] T, w w = [w w1... w wj ] T, D uw = diag([d i ]), D w = diag([d j ]), i C uw, j C w, A uw and A w is the matrix of I functions, A uw = I(E F uw), A w = I(E F w). E = [ ] T [ ] e 1 e 2... e Nnotch , with e i = i + L start 1 + 1, i = 1,..., N 2 notch. F uw = [ ] T [ ] fuw1 f uw2... f uwi, F w = [ ] T [ fw1 f w2... f wj ], with f uwi C uw, f wj C w. For those unweighted subcarriers, w uwi is 1 since there is no change to the original data.

16 Partial Weighting Mathematic Calculation Cont. The power spectrum can be show as y 2 = B uw w uw + B w w w 2 = (B uw w uw + B w w w ) (B uw w uw + B w w w ) = ((B uw w uw ) T + (B w w w ) T )(B uw w uw + B w w w ) = (w T uw BT uw + wt w BT w )(B uww uw + B w w w ) = P + Qw w + w T wrw w, with P = w T uwb T uwb uww uw; Q = (w T wb T wb w) + (B T wb uww uw) T ; R = B T uwb w.

17 Parameter Design Constraint of the Weights Let us take BPSK as an example. For BPSK, input data should be ± E. For the weighting factor w i = a i + b i, i A on constraints { 1 α1 < p i < 1 + α 1 α 2 < q i < α 2 are added to it to make sure the whole spectrum stay in almost the same place, with both α 1 and α 2 close to 0. Reason: Receiver can easily detect the data without knowing the weights. Keep the total power of the weighting factor as Len.

18 Parameter Design However, in this point, we do some approximation on the power constraint. Instead of making i A on ( p i 2 + q i 2 ) = N N notch, we simplify the constraint as i A on p i = N N notch. From Figure 2, it is easy to be found that the simplified constraint is tangent to the original one.

19 Parameter Design With the new power constraint, the problem finally becomes Minimize X T S T SX Subject to GX R and HX = l, where G = [ I I ], R = r 1 r 2 r 3 r 4, H = [ ] T, and l = N Nnotch, with r 1 = (1 + α 1 ) [ ] T, r 2 = α 2 [ ] T, r 3 = (α 1 1) [ ] T, r 4 = α 2 [ ] T. This is now Quadratic Programming problem, and can be solved by well known methods.

20 Parameter Design BER Calculation For the BER performance of the system, the power of bit (1 α 1 ) E Eb [(1 + α 1 ) 2 + α 2 2] E, the minimun distance between two different symbol 2 E 2α 1 E dmin 2 E + 2α 1 E The BER of BPSK for AWGN channel is dmin 2 P eawgn = Q( ). 2N 0 For a point with the position (x, y), d min = x + E(1 α 1 ), and E b = x 2 + y 2. z = E b x 2 + y 2 = dmin 2 [x + E(1 α 1 )] 2

21 Parameter Design BER of BPSK for AWGN channel 10 0 BER of BPSK for AWGN channel(α 1 =α 2 =0.3) Lower Bound for BPSK AWGN Upper Bound for BPSK AWGN BPSK AWGN Bit Error Rate E b /N 0 (db)

22 Simulation Result Environment Setting In the simulation, we consider an OFDM system with N = 64 subcarriers. The simulation environment can be divided into three different cases: Different Simulation Cases Narrow Notch: N notch from Subcarrier 29 to 31 Wide Notch: N notch from Subcarrier 29 to 38 Partial Weighting for Wide Notch: Weights Apply to Subcarrier 24:28 and 39:43

23 Simulation Result Power Spectrum Example 5 Power Spectrum of BPSK Signal in OFDM System Notch Depth/dB original power spectrum power spectrum with d1 = 0, d2 = 0.1 power spectrum with d1 = 0, d2 = 0.2 power spectrum with d1 = 0, d2 = 0.3 power spectrum with d1 = 0.1, d2 = 0 power spectrum with d1 = 0.2, d2 = 0 power spectrum with d1 = 0.3, d2 = 0 power spectrum with d1 = 0.1, d2 = 0.1 power spectrum with d1 = 0.1, d2 = 0.2 power spectrum with d1 = 0.1, d2 = 0.3 power spectrum with d1 = 0.2, d2 = 0.1 power spectrum with d1 = 0.2, d2 = 0.2 power spectrum with d1 = 0.2, d2 = 0.3 power spectrum with d1 = 0.3, d2 = 0.1 power spectrum with d1 = 0.3, d2 = 0.2 power spectrum with d1 = 0.3, d2 = Index of OFDM Subcarrier

24 10 Simulation Result Narrow Notch Without Normalization 0.3 Contour of Notch Depth given by α 1 & α Notch Depth of 3 Subcarrier All Subcarrier Weighting without Normalization α α 2 10

25 14 Simulation Result Narrow Notch With Normalization 0.3 Contour of Notch Depth given by α 1 & α 2 Notch Depth of 3 Subcarrier All Subcarrier Weighting with Normalization α α

26 Simulation Result Wide Notch With Normalization Contour of Notch Depth given by α 1 & α Contour of the notch depth given by α1 and α2 Points given the same BER performance as α1 = α α

27 9 9.5 Simulation Result Partial Weighting for Wide Notch 0.3 Notch Depth of 10 Suncarrier Notch with 10 Subcarrier Weighting/dB α α 2 9

28 Simulation Result BER Performance 10 0 BER result for Coded Signal with Weight BER result for Uncoded Signal with Weight 10 1 BitErrorProbability E b /N 0

29 System Model Description What is Sequence Selection? Instead of doing optimization, we can generate a set of different weights and choose the one with best performance. Different Generating Rules Weights: +/-1 Other Real Weights in a Certain Region Complex Weight: Phase Rotation In this case, we assume the receiver knows the weights correctly. Hence, it does not change the BER performance.

30 Simulation Result Notch Depth Result of Sequence Selection Notch Depth/dB Number of Different Data Sequences

31 Conclusion Complex Weighting vs. Real Weighting Give the same or better Notch Depth with small range of weights BER performance is more or less the same. All-Subcarrier Weighting vs. Partial Weighting Better BER performance Less Optimization Calculation Loss in Notch Depth

32 What Can We Do in the Future? Work on the BER performance Simulation Extend the weighting scheme from BPSK to other modulation QPSK 16QAM Find proper sequence set for Sequence Selection How to design the size of the set How to design the sequence in the set