An Optimal Filter for Signals with Time-Varying Cyclostationary Statistics Matt Carrick1 , Jeffrey H. Reed1 , fred harris2 1 Virginia Tech, Blacksburg, VA USA Email: {mcarrick, reedjh}@vt.edu 2 San Diego State University, San Diego, CA USA Email:
[email protected]
Abstract—This paper presents a filter for exploiting timevarying cyclostationary statistics. The motivating example is designing redundancies into an OFDM signal to reject in-band, strong, wideband interference. Combining the filter with error correcting codes gives a 10 dB gain over LDPC codes and more than a 1000x reduction in BER. This allows the creation of multicarrier waveforms that are robust, reliable and resistant to interference. The filter improves upon existing techniques thought the ability to exploit time-varying spectral redundancy, a capability unique to this filter. The design equations for the optimal filter weights are derived, showing the upper bound for the performance of the filter. The applications range from control channel protection for next generation commercial broadband to mission-critical communications.
I. Introduction This paper proposes a system for designing cyclostationary properties into an OFDM signal and develops an optimal filter for exploiting such time-varying cyclostationary statistics. Time-varying spectral redundancy is designed into an OFDM signal, and then exploited at the receiver through the proposed filter. The introduction of redundancy forms a trade-off, similar to an error correcting code, reducing the the spectral efficiency but increasing the waveform’s robustness and resilience to interference. The contributions of this paper are as follows: • New optimal filter for exploiting time-varying spectral redundancies • Method for building robust and resilient multicarrier waveforms • Improves 10 dB over soft decision LDPC codes • Trades a 1/4 loss in spectral efficiency for a 1000x improvement in BER in interference The creation of cyclostationary features within OFDM signals has been proposed by [1], [2] for network identification and cognitive radio applications. In [2], [3] the cyclostationarity of an OFDM signal is incorporated by repeating symbols in frequency and time, respectively. The creation and development of the FRESH filter is given in [4] with frequency domain and adaptive implementations being developed later [5], [6]. The use of FRESH filtering to exploit the spectral redundancy in OFDM signals induced by the cyclic prefix is proposed in [7], [8].
The FRESH filter this paper proposes is doubly timevarying. The FRESH filter of [4] is time-varying due to its periodic frequency shifters, but the filter weights themselves are located within time-invariant filters. This proposal includes the time-varying nature due to the frequency shifters, as well as applying sets of time-varying filter weights, analogous to the operation of a polyphase filter bank. As such, it will be referred to as a Time-Varying FRESH (TV-FRESH). This paper presents a method for rejecting interference through designing time-varying spectral redundancy into a multicarrier waveform and using the TV-FRESH filter to perform the optimal soft combining at the receiver. Timevarying spectral redundancy is created by repeating symbols in both time and frequency within an OFDM waveform. The proposed filter structure can coherently combine the redundant spectrum to form a signal processing gain and extract information even in the presence of strong in-band interference. A block digram describing this process is given in Figure 1. The method will be demonstrated for an Orthogonal Frequency Division Multiplexing (OFDM) waveform, but is applicable to any multicarrier waveform including Filter Bank Multicarrier (FBMC). The remainder of the paper is organized as follows. Section II provides background information with regards to cyclostationary signals and FRESH filtering. Section III introduces a TV-FRESH filter and describes an application which uses this filter structure. Section IV presents simulation results and Section V concludes the paper. II. Background A. Cyclostationary Signals Cyclostationary signals by definition are those which contain second order periodicity in the time domain and spectral redundancy in the frequency domain. The spectral correlation density function, S αx ( f ) = lim T T →∞
Z
T 2
− T2
α ∗ α XT t, f + XT t, f − dt, 2 2
(1)
is a measure of spectral redundancy at cycle frequency α [4] and XT (t, f ) is the Fourier transform of the signal x(t),
+
Multicarrier Demodulator
W2
+
W3
+
Multicarrier Modulator
Symbol Estimates
W1
+
Repeated Symbols
+
Resource Mapping & Symbol Repetition
W0
+
Data Symbols
Interference
Multicarrier Waveform
TV-FRESH and Demodulator
Fig. 1: A system level diagram of the proposed approach showing the creation and demodulation of the OFDM signal with symbol repetition.
1 XT (t, f ) = T
Z
t+ T2
− j2π f τ
x(τ)e
dτ.
(2)
time-varying spectral redundancy. An OFDM signal with N subcarriers is represented in the time domain by [2]:
t− T2
A cycle frequency is a relative difference in frequency upon which the signal x(t) spectrally correlates, S αx ( f ) , 0, and there may be multiple cycle frequencies present in one signal. Cyclostationary signals may also have conjugate spectral correlation, S βxx∗ ( f ) , 0. The conjugate spectral correlation function is:
˜ = d(t)
L−1 X N−1 X
T →∞
Z
T 2
− T2
β β XT t, − f + dt. XT t, f + 2 2
(3)
The cycle frequencies of the conjugate spectral correlation in (3) are denoted by β to distinguish them from the α cycle frequencies of the spectral correlation in (1). Therefore a signal is cyclostationary for any combination of S αx ( f ) , 0 for α , 0 and S βxx∗ ( f ) , 0. B. FRESH Filtering A Frequency Shift (FRESH) filter is the optimal filter for cyclostationary signals [4]. It exploits spectrally redundant information in cyclostationary signals to improve the estimates of the desired signal. The process filters the received signal through a parallel set of frequency shifts followed by linear time-invariant filters and sums the result. The FRESH filter is defined by
and in the frequency domain by L−1 X N−1 X
˜ f) = D(
˘ = d(t)
ak (t) ⊗ x(t)e j2παk t +
k
X
bn (t) ⊗ x∗ (t)e j2πβn t ,
n − j2πT l e . al,n Q f − N n=0
(6)
In these models al,n is the symbol of the lth OFDM symbol transmitted on subcarrier n, and q(t) is the rectangular pulse shape, where Q( f ) = F {q(t)} and F {·} is the Fourier transform. T is the length of the OFDM symbol, including the cyclic prefix, where T = TCP + T s . This standard OFDM model is modified by incorporating symbol repetition across both time and frequency leading to block-based transmissions, where the number of sequential OFDM symbols that contain spectral redundancy in time is B. Consider the lth block of OFDM symbols for which there are M symbols to be transmitted, represented by al,0 , al,1 , . . . , al,M−1 . The mth symbol al,m is repeated R(m) times, and is mapped onto subcarrier frequency fm,r . The time and frequency domain representations of the signal model of the cth OFDM symbol within the lth block is given by: dl,c (t) =
M−1 X R(m)−1 X m=0
X
(5)
l=0 n=0
l=0
S βxx∗ ( f ) = lim T
n
al,n q (t − lT ) e j2π N t ,
al,m q(t − (Bl + c)T )e j2π fm,r t ,
(7)
r=0
(4)
n
where ⊗ is the convolution operator [4]. The Wiener filter is a degenerate case of the FRESH filter where α = 0, and in this case no spectral redundancy is exploited by the filter leading to the decrease in performance relative to the FRESH filter. III. Multicarrier Waveform and Demodulator
Dl,c ( f ) =
M−1 X R(m)−1 X m=0
al,m Q( f − fm,r ) · e− j2π( f − fm,r )(Bl+c)T .
The signals for all time are given by: d(t) =
B−1 XX
D( f ) =
dl,c (t),
(9)
c=0
l
A. Signal Model In this section we provide the standard OFDM signal model and then adapt it to include symbol repetition creating
(8)
r=0
B−1 XX l
c=0
Dl ( f ).
(10)
α1,1,p = 8/16 α2,1,k = 2/16 OFDM Symbol Index
repeated B/M times over all B OFDM symbols. To create the spectral redundancy a circular shift in frequency of bM/B subcarriers is applied to the bth OFDM symbol. An example of this pattern is given in Figure 2 which illustrates the pattern when B = 2, N = 8 and M = 4. The best repetition pattern to be used is subject to an optimization that is outside the scope of this paper.
α2,3,n = -2/16
1 2 3 4 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Subcarrier Index Fig. 2: Spectral redundancy is distributed across both time and frequency within this pattern of B = 4 OFDM symbols, N = 16 subcarriers and M = 8 data-symbols. Three cycle frequencies illustrate the time-varying spectral redundancy. The arrows indicate in which direction the frequency shift is performed.
The transmitted signal d(t) will be corrupted by cyclostationary interference i(t) and stationary white noise n(t), resulting in the time domain and frequency domain received signals x(t) = d(t) + i(t) + n(t) and X( f ) = D( f ) + I( f ) + N( f ). B. Cycle Frequencies The relationship between the repeated symbols in time and frequency forms the cycle frequencies. The spectral correlation forming the α and β cycle frequencies must be considered separately and is dependent on the modulation being used. For spectral correlation to exist E {υυ∗ } , 0 where υ is the set of all constellation points within the modulation [9], and this is true for Pulse Amplitude Modulation (PAM), Quadrature Amplitude Modulation (QAM) and Phase Shift Keying (PSK). For conjugate spectral correlation to exist E {υυ} , 0 which is true for BPSK and PAM modulations. The cycle frequencies corresponding to the repetitions of symbol al,m must be known to the receiver so the spectral redundancy can be exploited. The cycle frequencies are derived from the relative difference in frequency of repeated data symbols. The notation αc,b,u is used for the uth cycle frequency for which there is spectral correlation of the bth OFDM symbol with respect to the cth OFDM symbol. Figure 2 demonstrates examples of the notation, with α2,1,k aligning the kth spectral redundancy in OFDM symbol 1 with that of OFDM symbol 2. Uc,b represents the number of such cycle frequencies between OFDM symbol c and OFDM symbol b. Similarly βc,b,v is the vth cycle frequency of the conjugate spectral correlation of the bth OFDM symbol with respect to the cth OFDM symbol. Vc,b is the number of conjugate cycle frequencies between OFDM symbol c and OFDM symbol b. C. Repetition Pattern The repetition pattern in Figure 2 is an example of a simple yet very effective repetition technique that maximizes the temporal and frequency diversity of the repeated symbols. For N subcarriers and a block size of B OFDM symbols, there are M unique symbols to be transmitted, where NB/M is an integer. Within the frequency domain, the M symbols are
D. TV-FRESH Filter The Fast Fourier Transform (FFT) is used to demodulate the OFDM signal, transforming the time domain signal into frequency domain symbols after removal of the cyclic prefix. A TV-FRESH filter is employed in the frequency domain to exploit the time-varying spectral redundancy of the received signal. Combining these two operations forms the TV-FRESH structure which jointly filters each block of B symbols. The symbols are buffered in memory, allowing the spectral redundancy across multiple OFDM symbols to be exploited. The estimate of the cth OFDM symbol Dl,c is: c,b −1 B−1 UX X Gc,b,u ( f )Xl,b ( f − αc,b,u ) Dˆ l,c ( f ) = b=0
u=0
+
VX c,b −1 v=0
∗ Hc,b,v ( f )Xl,b (− f − βc,b,v ) ,
(11)
and the filtering structure is illustrated in Figure 3. The filter output Dˆ l,c ( f ) is the estimate of the cth OFDM symbol Dl,c ( f ). By setting B = 1, (11) collapses to a frequency domain implementation of the FRESH filter described in [4]. The estimate Dˆ l,c ( f ) must be computed for c = 0, 1, . . . , B − 1 to filter the entire block of OFDM symbols, however as in (7) only M unique symbols are transmitted per block. Therefore for practicality the TV-FRESH only needs to estimate a total of M symbols which reduces the complexity. E. MMSE Filter Coefficients The optimal filter weights (15) and (16) are those which produce the Minimum Mean Squared Error (MMSE) estimate of the desired signal at the output of the TV-FRESH. This assumes perfect knowledge of the transmitted signal which is not possible in practice but allows a limit to be achieved upon which other implementations can be compared against. An adaptive approach to setting the filter weights where the transmitted signal is unknown can be implemented as in [6]. To derive MMSE filter weights the derivative of the mean squared error, n o ∗ E MS E = E El,c ( f )El,c (f) , (12) is taken with respect to conjugate of the filters G and H separately: ∂E MS E = 0, ∂G∗c,p,k ( f )
(13)
∂E MS E = 0. ∗ ∂Hc,m,n (f)
(14)
...c = B-1
+ +
El,0(f)
...
G0,0,U0,0-1(f)
+
+
+ G0,0,0(f)
... Circ. Shift Kα0,0,U0,0-1
Dl,0(f) Dl,1(f)
...
Circ. Shift Kα0,0,0
+
b=0
+
Xl,0(f)
K-FFT
Dl,0(f)
...
...
x[k]
c=1 c=0
+
B OFDM Symbols Stored in (B) K Sample Buffers
Dl,B-1(f)
*
X (-f)
+ H0,0,V0,0-1(f)
Gc,1,0(f) to Gc,1,Uc,1-1(f), Hc,1,0(f) to Hc,1,Vc,1-1(f)
b=1
+
Xl,1(f)
...
Circ. Shift Kβ0,0,V0,0-1
K-FFT
H0,0,0(f)
...
...
Circ. Shift Kβ0,0,0
...
...
Xl,B-1(f) K-FFT
b = B-1
Gc,B-1,0(f) to Gc,B-1,Uc,B-1-1(f), Hc,B-1,0(f) to Hc,B-1,Vc,B-1-1(f)
Fig. 3: A block diagram of the TV-FRESH filter from (11) that exploits time-varying spectral redundancy.
α S dcc,p,k ,x p
c,b −1 B−1 UX αc,p,k X αc,p,k + αc,b,u α −αc,b,u f− = Gc,b,u ( f )S xcc,p,k f− ,xb 2 2 b=0 u=0 +
VX c,b −1
Hc,b,v ( f )
v=0
β S dcc,m,n ∗ ,xm
β −αc,p,k S xcc,b,v ,xb∗
βc,b,v + αc,p,k f− 2
!∗ #
c,b −1 ! X ! B−1 UX βc,m,n + αc,b,u βc,m,n β −αc,b,u Gc,b,u ( f )S xcc,m,n f − = f− ,xb∗ 2 2 b=0 u=0 +
VX c,b −1
β
Hc,b,v ( f ) S xcc,m,n ,xb
v=0
p = 0, 1, . . . , B − 1;
k = 0, 1, . . . , Uc,p − 1;
The frequency domain error, El,c ( f ) = Dl,c ( f ) − Dˆ l,c ( f ),
(17)
is then substituted into (13)-(14) to produce the the optimal filter design equations (15)-(16). IV. Simulation Results The performance of the proposed method is measured under heavy interference. The interference scenario is motivated by co-existence of satellite communications and 5G cellular [10]. The scenario simulation includes a Digital Video Broadcast Second Generation (DVB-S2)-like signal interfering with an OFDM signal.
m = 0, 1, . . . , B − 1;
−βc,b,v
βc,m,n + βc,b,v −f + 2
(15)
(16) !#
n = 0, 1, . . . , Vc,m − 1
The interference is a single carrier QPSK signal with roll-off factor α = 0.25, power Pi /N0 = 20 dB and covers 1/2 and 1/4 of the OFDM signal bandwidth in the two scenarios. The spectral redundancy of the cyclostationary interferer is exploited by incorporating its cycle frequencies [9], the cycle frequencies of the OFDM signal and the combinations of differences between the two sets into the TV-FRESH. In these results the OFDM signal is using QPSK on all 128 subcarriers. Symbol repetition is a form of bandwidth expansion thus comparisons are made against soft decision LDPC and soft decision convolutional codes for similar overhead rates. Figures 4a and 4b compare the bit error rates for the given interference scenarios for symbol repetition, LDPC and convolutional code rates of 1/2 and 1/4 as well as a hybrid approach using a 1/2 error
the TV-FRESH out perform the LDPC codes by 10 dB and convolutional codes by 4 dB. However, the combination of the symbol repetition and convolutional codes produces the best BER of all of the approaches, producing a BER better than 10−4 and 10−6 , a 1000x reduction in BER as compared to the next best method. The symbol repetition and TV-FRESH is able to reject a large amount of the interference because it exploits the spectral redundancy at the signal level, making use of the most amount of information the signal has prior to demodulation and decoding. The LDPC code performs very poorly due to the thresholding effect, as the receive SINR pushes the BER above the knee in the curve resulting in a very poor BER.
0
10
−1
10
−2
Bit Error Rate
10
−3
10
1/2 Symbol Rep 1/2 Conv 1/2 LDPC 1/4 Sym 1/4 Conv 1/4 LDPC 1/2 Symbol Rep, 1/2 Conv 1/2 Symbol Rep, 1/2 LDPC Wiener Filter AWGN Theo
−4
10
−5
10
−6
10
−7
10
0
2
4
V. Conclusion 6
Eb/N0 (dB)
8
10
(a) Bit error rate plot comparing the performance of the symbol repetition approach with LDPC and convolutional codes while in strong interference, with interference covering 1/2 the OFDM signal bandwidth. 0
10
−1
10
References
−2
Bit Error Rate
10
−3
10
1/2 Symbol Rep 1/2 Conv 1/2 LDPC 1/4 Sym 1/4 Conv 1/4 LDPC 1/2 Symbol Rep, 1/2 Conv 1/2 Symbol Rep, 1/2 LDPC Wiener Filter AWGN Theo
−4
10
−5
10
−6
10
−7
10
The concept of designing time-varying spectral redundancy into multicarrier waveforms was proposed, along with the filter structure to optimally combine the redundant symbols. The proposed method is unique in its ability to deal with harsh interference with very small overhead rates. Simulation results show that using both symbol repetition and error correcting codes can improve the BER by a factor of 1000x in the given interference scenarios, and shows a gain of 10 dB over soft decision LDPC codes.
0
2
4
6
Eb/N0 (dB)
8
10
(b) Bit error rate plot comparing the performance of the symbol repetition approach with LDPC and convolutional codes while in strong interference, with interference covering 1/4 the OFDM signal bandwidth.
correcting code and 1/2 symbol repetition. To incorporate time diversity the spectral correlation is distributed across B = 2 OFDM symbols when the repetition rate is 1/2 and B = 4 when the repetition rate is 1/4. The Wiener filter is also applied when convolutional codes are used to imitate a typical OFDM receiver. Figures 4a and 4b show that the proposed method of symbol repetition and TV-FRESH outperforms traditional OFDM receivers using LDPC codes and convolutional codes under the interference scenarios. The symbol repetition method and
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