Performance Evaluation of Quantization Effects on Multicarrier Modulated Signals Teresa Arau´ jo and Rui Dinis, Member, IEEE ISR-Instituto Superior Te´ cnico, 1049-001 Lisboa, Portugal Phone: + 351 21 8418298, Fax: + 351 21 8418290, e-mail: ftaraujo,
[email protected]
Abstract The numerical accuracy in the DFT/IDFT operations can have a significant impact on MC (MultiCarrier) modulated signals. This accuracy can be modeled by including appropriate quantization devices in the transmission chain. In this paper, we present an analytical approach for analyzing the impact of the quantization effects in MC signals. For this propose, we characterize statistically the signal along the transmission chain, taking advantage of the Gaussian behavior of MC signals with a high number of subcarriers and employing wellknown results on Gaussian signals and memoryless nonlinearities. This statistical characterization can then be used for performance evaluation of given quantization characteristics, in a simple and computationally efficient way, as well as to its optimization. Keywords: Multicarrier modulations, quantization, nonlinear effects, Gaussian processes I. Introduction In recent years, MC modulations schemes (MultiCarrier) have been selected for several digital transmission systems: the OFDM schemes (Orthogonal Frequency Division Multiplexing) [1] were adopted for both digital broadcasting systems and wireless LANs (Local Area Network) [2] and the DMT schemes (Discrete MultiTone) were adopted by the ADSL standard (Asymmetric Digital Subscription Line) [3]. One of the main reasons for the high interest behind these MC modulations is their ability to cope with severe time-dispersive channels without requiring complex receivers, thanks to its FFT-based (Fast Fourier Transform) implementations. However, MC signals have high envelope fluctuations, making them very prone to nonlinear distortion effects. When the number of subcarriers is high, MC signals exhibit a Gaussianlike behavior, which can be used for a theoretical evaluation of nonlinear effects [4]-[8]. The clipping of a DMT signal was considered in [4]. The impact of bandpass memoryless nonlinear devices [9] on OFDM signals was considered in [5]-[7]. The analytical approach of [4] was extended to the so-called I-Q memoryless nonlinear devices (i.e., nonlinear devices operating separately in the real and imaginary parts of the complex envelope of the OFDM signal) in [8]. Another problem associated to MC signals is the numerical accuracy required in the DFT/IDFT operations, which can have
a significant impact on the MC transmission performance, especially when large constellations are employed. This accuracy can be modeled as appropriate quantization effects associated to the input and/or the output of each DFT/IDFT computation. The evaluation of quantization effects is a well-known problem in ADC (Analog-to-Digital Conversion). The usual approach is to assume that an uniformly-distributed noise is added to the quantized signal [10], [11]. However, this approach is not suitable if there are clipping effects (i.e., the ”saturation” of the quantizer is not a very rare event) and/or for nonuniform quantizers. Clipping effects on Gaussian noise were studied in [12] and a general theory for non-uniform quantizers can be found in [13]. An analytical approach for evaluating the impact of memoryless nonlinear devices in real-valued Gaussian signals was presented in [14] (and, more recently, in [15]) and used for evaluating quantization and clipping effects at the multicarrier receiver. The impact of the oversampling factor on the SIR levels was also considered there. In this paper, we study the quantization effects on the complex envelope of MC signals. These quantization effects occur at both the transmitter and the receiver and are associated to the numerical accuracy of the DFT computations. For this purpose, we include an appropriate statistical characterization of the signal along the transmission chain which takes advantage of the Gaussian behavior of the complex envelope of MC signals with a high number of subcarriers and employs well-known results on Gaussian signals and memoryless nonlinearities. This statistical characterization can then be used for performance evaluation of given quantization characteristics, in a simple and computationally efficient way, as well as to its optimization. This paper is organized as follows. In sec. II we present the quantization effects, inherent to the accuracy of the DFT/IDFT operations, on MC signals. Sec. III presents an analytical, statistical characterization of the transmitted and received signals, taking into account these quantization effects. In sec. IV we present some numerical results and sec. V is concerned with the conclusions and final remarks of this paper. II. Quantization Effects on Multicarrier Modulated Signals Fig. 1.A presents the transmission chain considered for MC modulations. Each DFT/IDFT operation is modeled as an ideal DFT/IDFT operation (i.e., with infinite precision), followed and preceeded by quantization blocks, modeling numerical
{S k }
{s }
{sn } IDFT
Q n
Q1
{ν n } Channel {Hk}
{Hˆ }
A
k
{Y }
{y }
{yn } +
Q k
Q n
DFT
Q2
{Sˆ } k
Dec.
B Re{.}
gQi(.)
x
+ Im{.}
gQi(.)
xQ
X j
Fig. 1.
Multicarrier transmission chain (A) and detail of the quantizer block Qi (B).
accuracy issues and clipping effects (see [18])1 . However, as shown in fig. 1.A, we will ignore the quantization effects on the frequency-domain samples (i.e, at the input of the ”ideal IDFT device” and the output of the ”ideal DFT device”). This can be justified as follows: ²
²
²
The quantization effects in the frequency-domain samples have a local effect, restricted to a given subcarrier. On the other hand, since the quantization effects in the timedomain samples produce spectral widening (as it will be shown in the following), the resulting nonlinear distortion effects can affect all subcarriers. The frequency-domain signals to be transmitted have lower dynamic range than the corresponding time-domain signals (e.g., belonging to a given PSK (Phase Shifted Keying) or QAM (Quadrature Amplitude Modulation) constellation, instead of having a quasi-Gaussian distribution). The received samples are usually submitted to a decision device (which can be modeled as a suitable quantization characteristic).
Each of the N frequency-domain symbols to be transmitted is selected from a given constellation, according to the transmitted data and an augmented block is formed by adding N 0 ¡ N idle subcarriers, i.e., with Sk = 0. As shown in fig. 2, the complex envelope of the bandpass signal is referred to the frequency fC = f0 + ¢N F (which corresponds to the zero frequency of the complex envelope), where f0 is the central frequency of the pass-band signal and F denotes the subcarrier separation. For conventional MC implementations, the complex envelopes are referred to the central frequency of the spectrum, leading to fC = f0 and ¢N = 0. It will be shown in the following that the selected ¢N can have a significant effect on the quantizers’ performance. An ”ideal IDFT device” produces the time-domain block fsn ; n = 0; 1; : : : ; N 0 ¡1g = IDFT fSk ; k = 0; 1; : : : ; N 0 ¡1g, which can be regarded as a sampled version of the MC 1 Basically, this means that the DFT operation is performed ideally (without numerical errors) and the effect of the numerical errors is simply included by ignoring the least reliable bits.
0
burst, with an oversampling factor MT x = NN , followed by a quantization operation (which includes clipping effects), leading to the block of time-domain samples to be transmit0 ted fsQ n ; n = 0; 1; : : : ; N ¡ 1g. The quantizer operates on complex-valued symbols and can be regarded as an ”I-Q” memoryless nonlinearity, which separately operates on the real and the imaginary parts of each complex sample x [8]. This is done in accordance with (see fig. 1.B) sQ n = gQ1 (Refsn g) + jgQ1 (Imfsn g) ;
(1)
where gQ1 (x) denotes an appropriate odd function. When the guard interval is longer than the length of the overall channel impulse response, the received time-domain samples, yn , are such that fyn ; n = 0; 1; : : : ; N 0 ¡ 1g = IDFT fYk ; k = 0; 1; : : : ; N 0 ¡ 1g, where the received symbol for the kth subcarrier is Yk = Hk SkQ + Nk ;
(2)
with Hk and Nk denoting the corresponding channel frequency response and channel noise, respectively, and fSkQ ; k = 0 0; 1; : : : ; N 0 ¡ 1g = DFT fsQ n ; n = 0; 1; : : : ; N ¡ 1g. The samples yn are quantized before the DFT operation, leading to the samples ynQ = gQ2 (Refyn g) + jgQ2 (Imfyn g) ;
(3)
also with an appropriate odd function gQ2 (x). The DFT operation produces the frequency-domain block fYkQ ; k = 0; 1; : : : ; N 0 ¡ 1g, which is used for the detection of the transmitted symbols. If we consider floating-point operations, the numerical representation of a given x takes the form xQ = §m(x)2¡e(x) sM ;
(4)
where sM denotes the saturation level (i.e., the clipping effect inherent to the quantization characteristic) and m(x) and e(x) have Nm and Ne bits, respectively (by including the sign bit, the number of bits required to represent x is then Nt = Ne + Nm + 1). The corresponding binary representations are (e) (e) (e) (m) (m) (m) e(x) = [b1 b2 ¢ ¢ ¢ bNe ]2 and m(x) = [0:b1 b2 ¢ ¢ ¢ bNm ]2
(e)
(m)
where bi and bi are the ith bits of m(x) and e(x), respectively. To avoid multiple binary representations of a given x, it is assumed that [00 ¢ ¢ ¢ 0]2 · e(x) · [11 ¢ ¢ ¢ 1]2 (i.e., 0 · e(x) · 2Ne ¡ 1) and [0:00 ¢ ¢ ¢ 0]2 · m(x) · [0:11 ¢ ¢ ¢ 1]2 if e(x) = [11 ¢ ¢ ¢ 1]2 and [0:10 ¢ ¢ ¢ 0]2 · m(x) · [0:11 ¢ ¢ ¢ 1]2 , otherwise (i.e., 0 · m(x) · 1 ¡ 2¡Nm if e(x) = 2Ne ¡ 1 and 2¡1 · m(x) · 1 ¡ 2¡Nm , otherwise). Fig. 3 presents the evolution of the quantization characteristics, gQ (¢), for (Nm ; Ne ) = (5; 0) (uniform quantization) and (Nm ; Ne ) = (3; 2) (non-uniform quantization). It should be noted that, with this representation, the number of quantization levels is 2Nt for Ne = 0 and 2Nt ¡1 + 2Nm for Ne 6= 0. A
as samples of a zero-mean complex Gaussian process. If E[Sk Sk¤0 ] = Gs (k)±k;k0 (±k;k0 = 1 for k = k 0 and 0 otherwise)2 , then E[sn ] = 0 and E [sn s¤n0 ] = Rs (n ¡ n0 ) = ¶ µ NX ¡1 NX ¡1 1 kn ¡ k 0 n0 ¤ = = E [Sk Sk0 ] exp j2¼ (N 0 )2 N0 k=0 k0 =0 0 ¶ µ ¡1 NX ¤ £ 1 k(n ¡ n0 ) 2 = E jSk j exp j2¼ (N 0 )2 N0 0
0
(5)
k=0
(n; n0 = 0; 1; : : : ; N 0 ¡ 1), i.e., the autocorrelation of the samples sn is fRs (n); n = 0; 1; : : : ; N 0 ¡ 1g = IDFT fGs (k)=N 0 ; k = 0; 1; : : : ; N 0 ¡ 1g. The variance of both Refsn g and Imfsn g is 0
-fC
0
-f0
f0
∆N ⋅ F
fC
f
∆N ⋅ F
B
C
{S k } ∆N 0
N '-1
0
f
¾2 =
−∆N ⋅ F
In the following, we take advantage of the quasi-Gaussian nature of the samples sn for obtaining the statistical characterization of the transmitted blocks. It is well-known that the output of a memoryless nonlinear device with a Gaussian input can be written as the sum of two uncorrelated components: a useful one, proportional to k the input, and a self-interference one [16]. Since the real and imaginary parts of the quantizer input are submitted to identical memoryless nonlinearities, its output can be written as Q1 sQ sn + dQ1 n =® n
where
0.8
g (x)/s
0.6
Q
0.6
0.4
0.4
0.2
0.2
0
0
0.2
0.4
0.6 x
0.8
1
0
0
0.2
0.4
0.6 x
= 0 and
(7)
¶ µ x2 xgQ1 (x)exp ¡ 2 dx: 2¾ ¡1 (8) The average power of the useful component is S Q1 = j®Q1 j2 ¾ 2 , and the average power of the self-interference Q1 Q1 ¡S Q1 , where Pout denotes component is given by I Q1 = Pout the average power of the signal at the nonlinearity output, given by ¶ µ Z +1 1 x2 Q1 2 2 gQ1 (x)exp ¡ 2 dx: Pout = E[gQ1 (x)] = p 2¾ 2¼¾ ¡1 (9) It can be shown (see Appendix A) that the autocorrelation of Q¤ the output samples, RsQ (n¡n0 ) = E[sQ n sn0 ], can be expressed as a function of the autocorrelation of the input samples in the following way:
M
0.8
gQ(x)/sM
1
E[sn dQ1¤ n0 ]
®Q1 =
B
1
(6)
k=0
Fig. 2. PSD of the bandpass signal (A) and the complex envelope referred to the frequency fC (B), as well as the corresponding block fSk ; k = 0; 1; : : : ; N 0 ¡ 1g when N = 9, N 0 = 16 and ¢N = 3 (C).
A
N ¡1 1 1 X E[jSk j2 ]: Rs (0) = 2 2N 2
0.8
E[xgQ1 (x)] 1 = p E[x2 ] 2¼¾ 3
1
Fig. 3. Evolution of gQ (¢) for (Nm ; Ne ) = (5; 0) (A) or (Nm ; Ne ) = (3; 2) (B).
III. Characterization of the Transmitted and Received Signals When the number of subcarriers is high (N >> 1) the time-domain coefficients sn can be approximately regarded
RsQ (n ¡ n0 ) =
+1 X °=0
Z
Q1 2P2°+1
+1
f2°+1 (Rs (n ¡ n0 )) 2°+1
(Rs (0))
(10)
where Rs (n ¡ n0 ) is given by (5), f2°+1 (x) is given by (43) Q1 and the coefficient P2°+1 , denoting the total power associated 2 For OFDM schemes, it is usually assumed that E[jS j2 ] = G (k) s k is constant in-band and zero out-of-band. For DMT schemes with loading, 2 E[jSk j ] = Gs (k) is not necessarily constant in-band, and can even be zero for subcarriers at frequency notches
to the IMP (Inter-Modulation Product) of order 2° + 1, can be computed as described in Appendix A. Since RsQ (n ¡ n0 ) = j®j2 Rs (n ¡ n0 ) + E[dn d¤n0 ];
(11)
it can be easily recognized that P1Q1 = j®Q1 j2 ¾ 2 , with 2¾ 2 = Q1¤ (n¡n0 ) = E[dQ1 E[jsn j2 ], and RdQ1P n dn0 ] is obtained by using P 1 1 °=1 instead of °=0 in the right-hand side of (10). Having in mind (7), the frequency-domain block fSkQ ; k = 0 0; 1; : : : ; N 0 ¡ 1g = DFT fsQ n ; n = 0; 1; : : : ; N ¡ 1g can obviously be decomposed into useful and self-interference components, i.e., SkQ = ®Q1 Sk + DkQ1 , where fDkQ1 ; k = 0 0; 1; : : : ; N 0 ¡1g denotes the DFT of fdQ1 n ; n = 0; 1; : : : ; N ¡ 1g. If gQ1 (¡x) = ¡gQ1 (x) (i.e., gQ1 (x) is an odd function of Q1 x), it can be shown that E[dQ1 n ] = 0, leading to E[Dk ] = 0. Moreover, i h = E DkQ1 DkQ1¤ 0 0 0 ¶ µ NX ¡1 NX ¡1 i h kn ¡ k 0 n0 Q1 Q1¤ = = E dn dn0 exp ¡j2¼ N0 n=0 0 n =0
0 = N 0 GQ1 (12) d (k)±k;k
0 (k; k 0 = 0; 1; : : : ; N 0 ¡1), where fGQ1 d (k); k = 0; 1; : : : ; N ¡ Q1 0 1g denotes the DFT of the block fRd (n); n = 0; 1; : : : ; N ¡ 1g. This means that the self-interference components associated to different subcarriers are uncorrelated. Similarly, Q E[SkQ SkQ¤ 0 ] = N Gs (k)±k;k0
(13)
Q1 2 0 where fGQ j Gs (k) + GQ1 s (k) = j® d (k); k = 0; 1; : : : ; N ¡ Q 0 1g denotes the DFT of fRs (n); k = 0; 1; : : : ; N ¡ 1g, given by (10). Let us consider now the transmission of the MC signal over a time-dispersive channel. From (2), we have Q1 yn = sQ1 sn + dQ1 n ¤ hn + ºn = (® n ) ¤ hn + ºn =
=
0 NX ¡1
n0 =0
(®Q1 sn¡n0 + dQ1 n¡n0 )hn0 + ºn ;
(14)
where ºn are the time-domain noise samples and fhn ; n = 0; 1; : : : ; N 0 32 ¡ 1g = IDFT fHk ; k = 0; 1; : : : ; N 0 ¡ 1g is the channel impulse response of the equivalent discrete channel (as usual, it is assumed that hn = 0 for n > NG with NG denoting the number of samples in the guard interval). Since the self-interference component dQ1 n is not Gausssian, yn is not Gaussian in the general case. However, if the channel has a large number of multipath components (as in severely timedispersive channels with rich multipath propagation), hn has a large number of non-zero terms and the samples yn can still be regarded as samples of a zero-mean Gaussian process3 . The corresponding autocorrelation is E[yn yn¤ 0 ] = Ry (n ¡ n0 ), 3 In fact, since the power of the self-interference component is typically much lower than the power of the useful component, the Gaussian approximation of the received samples can be very accurate, even for non-dispersive channels.
where fRy (n); n = 0; 1; : : : ; N 0 ¡ 1g = N1 IDFTfGy (k); k = 0; 1; : : : ; N 0 ¡ 1g, with Gy (k) = E[jYk j2 ] = jHk j2 GQ s (k) + E[jNk j2 ]. This means that the samples ynQ can be decomposed into uncorrelated useful and self-interference components as in (7), i.e., (15) ynQ = ®Q2 yn + dQ2 n ; with ®Q2 given by (8), with gQ1 replaced by gQ2 . The 0 Q Q¤ corresponding autocorrelation RQ y (n ¡ n ) = E[yn yn0 ] can be written as in (10), i.e., 0 RQ y (n ¡ n ) =
+1 X
Q2 2P2°+1
f2°+1 (Ry (n ¡ n0 ))
°=0
(Ry (0))
2°+1
;
(16)
Q2 denoting the total power once again, with the coefficient P2°+1 associated to the IMP of order 2° + 1. The corresponding frequency-domain samples YkQ can also be decomposed into useful and self-interference components, YkQ = ®Q2 Yk +DkQ2 , Q and E[YkQ YkQ¤ 0 ] = N Gy (k)±k;k 0 . By employing the statistical characterization of the frequency-domain block to be transmitted, one can calculate an ”equivalent signal to noise plus self-interference” for each subcarrier, given by
=
j®
Q1 Q2
®
2
2
Hk j E[jSk j ]
ESN Rk =
j®Q2 Hk j2 E[jDkQ1 j2 ] + E[jDkQ2 j2 ] + j®Q2 j2 E[jNk j2 ]
: (17)
It was observed that, when the number of active subcarriers is high enough to validate the Gaussian approximation for the time-domain samples at the nonlinearity input (say, N ¸ 64), our modeling approach is quite accurate. Moreover, under this ”high number of subcarriers” assumption, the selfinterference terms DkQ1 and DkQ2 typically exhibit quasiGaussian characteristics for any k, although, as stated above, Q2 the self-interference components dQ1 n and dn are obviously not Gaussian [7], [8]. The Gaussian approximation of the frequency-domain self-interference terms, DkQ1 and DkQ2 , is very accurate, unless very mild nonlinear characteristics are considered (in that case, almost all of the self-interference samples in the time-domain are zero and the corresponding frequency-domain samples are no longer Gaussian [17]). This means that the BER performance for the kth subcarrier can be easily obtained from the corresponding ESN Rk : for instance, if a QPSK constellation is considered, the BER for the kth subcarrier is ´ ³p Pb;k = Q (18) ESNRk ; with Q(¢) denoting the well-known Gaussian error function. Our simulations showed that this is a very accurate approximation for the BER performance. This method for statistical characterization of the transmitted blocks is quite appropriate whenever the power series in (10) and (16) can be reasonably truncated while ensuring an accurate computation. However, for strongly nonlinear conditions, such as the ones inherent to quantization characteristics, the
45
required number of terms becomes very high. In such cases, one can simplify the computation as explained below. When ° >> 1, ¾2°+1 ½ Rs (n ¡ n0 ) ¼0 (19) Im Rs (0)
35
SIRTx(dB)
and µ ¶2°+1 ¾2°+1 ½ Rs (n ¡ n0 ) Rs (n ¡ n0 ) ¼ Re ¼ ±n;n0 ; (20) Rs (0) Rs (0)
40
25
which means that the frequency-domain distribution of the power associated to a given IMP, fGs;2°+1 (k); k = 0; 1; : : : ; N 0 ¡o 1g = DFT n (Rs (n)=Rs (0))2°+1 ; n = 0; 1; : : : ; N 0 ¡ 1 , is almost constant, leading to Gs;2°+1 (k)=Gs;2°+1 (0) ¼ 1, and the contribution of the (2° + 1)th IMP to the output autoQ1 ±n;n0 . Therefore, correlation can be approximated by 2P2°+1
¼2
°X max
30
(Nm,Ne)=(6,0): ⋅ ⋅ ⋅ (N ,N )=(7,0): ___ m e (N ,N )=(8,0): − − − m e
20
15 2
Fig. 4.
2.5
3
3.5
4 sM/σ
4.5
5
5.5
6
Impact of sM =¾ on SIRT x , when ¢N = 0 and MT x = 1.
RsQ (n ¡ n0 ) ¼ Q1 P2°+1
f2°+1 (Rs (n ¡ n0 )) 2°+1
(Rs (0))
°=0
Q1;1 + 2P2° ± 0 (21) max +1 n;n
(°max >> 1), with Q1;1 P2° max +1
=
+1 X °=°max +1
Q1 P2°+1
=I
Q1
¡
°X max
Q1 P2°+1 ;
(22)
°=1
where I Q1 denotes the average power of the self-interference component. This means that, besides the computation of I Q1 = Q1 ¡ S Q1 , we just have to calculate the terms corresponding Pout to the first °max IMPs (°max = 10 or 20 is enough for most cases). A similar approach could be employed in the computation of RQ y (n).
larger sM =¾ the higher the ”step-size” of the quantizer, leading to worse SIRT x levels). Fig. 5 shows the impact of the oversampling factor MT x on SIRT x , for an uniform quantizer. From this figure, we can observe improvements on the SIRT x levels when we increase the oversampling factor MT x , especially for moderate and high values of sM =¾. These improvements are a consequence of the decreased aliasing effects in the in-band region when we increase the oversampling factor. It should be pointed out that the behavior of the SIR at the transmitter depicted in figs. 4 and 5 is similar to the behavior of the SIR at the receiver when the impact of the channel is not considered (as in [14]). 45
IV. Performance Results 40
35
SIRTx (dB)
In this section, we present a set of performance results concerning the quantization effects on MC signals. We consider an MC modulation with N = 256 active subcarriers (similar results were obtained for other values of N, provided that N >> 1), with the same attributed power, and the quantization/clipping characteristics described in sec. II. Fig. 4 shows the impact of the normalized saturation level, sM =¾ on the signal-to-interference ratio of the transmitted signals, SIRT x , defined as the average over the in-band subcarriers of j®Q1 j2 E[jSk j2 ]=E[jDkQ1 j2 ], for an uniform quantizer (Ne = 0), when ¢N = 0 (i.e., when the complex envelope of the MC signal is referred to the central frequency of the spectrum) and MT x = 1 (i.e., there is no oversampling). Clearly, there is an optimum value of sM =¾, that increases with the number of quantization bits. For small values of sM =¾ the quantization noise is mainly a consequence of saturation effects and SIRT x is almost independent of the number of quantization bits; for larger values of sM =¾, we almost do not have saturation effects and there is an improvement of 6dB on SIRT x for each additional quantization bit (naturally, the
MTx=1: ⋅ ⋅ ⋅ M =2: ___ Tx M =4: − − − Tx
30
25
20
15 2
2.5
3
3.5
4 sM/σ
4.5
5
5.5
6
Fig. 5. Impact of sM =¾ on SIRT x , when ¢N = 0 and (Nm ; Ne ) = (7; 0).
In fig. 6 we consider a non-uniform quantizer (i.e., Ne 6= 0). Although the non-uniform quantizers have good SIRT x levels for a wider range of saturation levels, for the optimum value
1
of sM =¾ the performance of a nonlinear quantizer is similar, or even better.
0.9 45
In−band 0.8
SIRTx (dB)
k
E[|D Q1|2]
40
35
0.6
20 2
___
(N ,N )=(7,0): ___ m e (N ,N )=(6,1): − − − m e (N ,N )=(5,2): ⋅ ⋅ ⋅ m e (Nm,Ne)=(4,3): − ⋅ −
30
25
2.5
3
3.5
4 sM/σ
4.5
0.4 −2
5
5.5
6
Fig. 7 shows the gains on the SIRT x levels associated with ¢N 6= 0. By adopting ¢N ¼ N=2 (i.e., by referring the complex envelope of the MC signal to a frequency on the edge of the ”useful” band), these gains are approximately 1dB for MT x = 4 and 2dB for MT x = 8. The SIR gains when ¢N 6= 0 can be explained in the following way: although the total self-interference power is independent of ¢N , when MT x is high enough the adoption of ¢N 6= 0 can lead to a decrease on the in-band self-interference levels and an increase on the out-of-band radiation levels (see fig. 8), leading to higher SIR values. 2.5
gains (dB)
2
Tx
∆N=0: ∆N=N/2: − − −
0.5
Fig. 6. Impact of sM =¾ on SIRT x , for non-uniform quantizers, when ¢N = 0 and MT x = 2.
SIR
0.7
1.5
−1.5
−1
−0.5
0 (k−∆N)/N
0.5
1
1.5
2
Fig. 8. Normalized values of E[jDkQ1 j2 ] for ¢N = 0 or ¢N = 1=2, when sM =¾ = 3:8, (Nm ; Ne ) = (7; 0) and MT x = 4.
tion level, the self-interference term associated to gQ1 (¢) is E[jDkQ1 Hk j2 ], which follows the evolution of the channel amplitude response. However, the self-interference term associated to gQ2 (¢), E[jDkQ2 j2 ], is almost independent of jHk j2 . This means that the later can have a significant effect on ESNRk for the frequencies corresponding to deep fades, leading to higher quantization requirements at the receiver (for this reason, a higher number of quantization bits is assumed at the receiver). This behavior is depicted in fig. 9, where we present the evolution of the analytical values of E[jDkQ2 j2 ] when a severe time-dispersive channel is consider. For the sake of comparisons, we also include values of E[jDkQ2 j2 ] obtained by simulation, slightly worse, but very close to the theoretical ones, with differences below 0.2dB. The corresponding values of ESN Rk are depicted in fig. 10, together with the ESN Rk values for an ideal AWGN channel (for the same quantization characteristics and SNR). Since ESNRk closely follows the evolution of jHk j2 (see fig. 9), it can have large fluctuations for frequency selective channels, with very poor values for frequencies in deep fades (this in not the case of the ideal AWGN channel, or flat fading channels, where ESN Rk has only minor fluctuations in the inband region).
1
V. Conclusions and Final Remarks M =2: − − − Tx M =4: ___ Tx M =8: ⋅ ⋅ ⋅
0.5
Tx
0 0
0.5
1
1.5
2
∆N/N
Fig. 7. Gains on SIRT x levels when ¢N 6= 0, for (Nm ; Ne ) = (7; 0) and MT x = 2.
Let us consider now a severely frequency-selective channel (the typical situation for MC transmission). At the detec-
An analytical approach for analyzing the impact of the numerical accuracy in the DFT computations was presented. This approach takes advantage of the Gaussian behavior of MC signals with a high number of subcarriers and can be used for the performance evaluation of a given quantization characteristic, in a simple and computationally efficient way, as well as its optimization. Our results indicate that the quantization requirements are higher at the receiver than at the transmitter, especially for severely frequency-selective channels. It is also shown that the selection of the oversampling factor and the ¢N factor can
A
Re{.}
10
g(.)
xin (t )
0
|Hk|2(dB)
xI (t )
yout (t )
−10
Im{.}
xQ (t )
−20
x
g(.) j
−30 −0.5
0
0.5
Fig. 11.
I-Q memoryless nonlinearity.
B
2
E[|DQ2 | ](dB) k
−45.6 −45.8
To calculate the output auto-correlation, we will proceed in the following way. The input complex envelope, xin (t) is modeled as a zero-mean, noise-like, Gaussian process. Clearly, the autocorrelation of xin (t) is given by
−46 −46.2 ___
−46.4 −46.6 −0.5
Theory: Simulation: − − − 0 k/N
Rin (¿ ) = E[xin (t)x¤in (t ¡ ¿ )] =
0.5
Fig. 9. Evolution of jHk j2 (A) and theoretical (solid line) and simulated (dotted line) values of E[jDkQ2 j2 ] (B), when the channel SNR is 40dB, gQ1 (¢) is characterized by (Nm ; Ne ) = (7; 0) and sM =¾ = 3:8 and gQ2 (¢) is characterized by (Nm ; Ne ) = (8; 0) and sM =¾ = 4:0.
= RII (¿ ) + RQQ (¿ ) + jRQI (¿ ) ¡ jRIQ (¿ ) with ¢
RII (¿ ) = E[xI (t)xI (t ¡ ¿ )]
(24)
RQQ (¿ ) = E[xQ (t)xQ (t ¡ ¿ )] = RII (¿ )
(25)
¢
45
____
¢
: Frequency−selective channel − − − : AWGN channel
(26)
RIQ (¿ ) = E[xI (t)xQ (t ¡ ¿ )] = ¡RQI (¿ );
(27)
where xI (t) = Refxin (t)g and xQ (t) = Imfxin (t)g. This means that, Rin (¿ ) = 2RII (¿ ) + j2RQI (¿ ) and, since RII (0) = RQQ (0) = ¾ 2 and RIQ (0) = RIQ (0) = 0, Rin (0) = 2RRR (0) = 2¾ 2 . The output complex envelope is given by yout (t) = yI (t) + jyQ (t) = g(xI (t)) + jg(xQ (t)) and its autocorrelation can be written as
35
ESNRk (dB)
RQI (¿ ) = E[xQ (t)xI (t ¡ ¿ )] ¢
40
30
25
¤ (t ¡ ¿ )] = Rout (t) = E[yout (t)yout
20
15 −0.5
(23)
0 k/N
0.5
Fig. 10. Evolution of ESN Rk for the frequency-selective channel of fig. 9, as well as an ideal AWGN channel.
have a significant impact on the SIR levels. It should be noted that our statistical approach could be employed with other I-Q memoryless nonlinearities such as an I-Q clipping for reducing the envelope fluctuations of the transmitted signals [8]. Therefore, there are also gains with the oversampling factor and the ¢N factor for that case. Appendix Consider an I-Q memoryless nonlinearity, with a bandpass input, where the real and imaginary parts of the input complex envelope (i.e., the ”in-phase” and ”quadrature” components) are submitted, separately, to two identical memoryless nonlinearities (see fig. 11).
= E[g(xI (t))g(xI (t ¡ ¿ ))] + E[g(xQ (t))g(xQ (t ¡ ¿ ))] + +jE[g(xQ (t))g(xI (t ¡ ¿ ))] ¡ jE[g(xI (t))g(xQ ((t ¡ ¿ ))] (28) It should be noted that the four expected values of the last equality of (28) involve two, jointly Gaussian random variables, submitted to two, identical nonlinearities g(x). This means that they have the form Z =
+1 ¡1
where
Z
+1 ¡1
Rab (¿ ) = E[g(xa )g(xb )] = g(xa )g(xb )p(xa ; xb )dxa dxb
(29)
¶ µ 2 1 xa + x2b ¡ 2½xa xb p exp ¡ 2¾ 2 (1 ¡ ½2ab ) 2¼¾ 2 1 ¡ ½2ab (30) denotes the joint probability density function of xa and xb , and E[xa xb ] ¢ (31) ½ab = p E[jxa j2 ]E[jxb j2 ] p(xa ; xb ) =
is the cross correlation coefficient for the random variables xa and xb . The direct computation of (29) is difficult since it requires the evaluation of a double integral for each value of ¿ . However, this computation can be made very simple. In fact, by using Mehler’s formula [19], we have µ µ ¶ ¶ +1 X ½nab xa xb p p H ; H p(xa ; xb ) = p(xa )p(xb ) n n 2n n! 2¾ 2¾ n=0 (32) where Hn (x) denotes an Hermite polynomial of degree n, defined as ª dn © (33) Hn (x) = (¡1)n exp(x2 ) n exp(¡x2 ) dx (see [20]). By using (32) in (29), and noting that the Hermite polynomials are even for n even and odd for n odd, we get +1 X
½2°+1 ab ¢ 22°+1 (2° + 1)! °=0 µ ¶ ¶2 µZ +1 x dx = g(x)p(x)H2°+1 p ¢ 2¾ ¡1 +1 X P2°+1 ½2°+1 ; = ab Rab (¿ ) =
In (34) P2°+1 denotes the total power associated to the IMP of order 2° + 1, given by 1 P2°+1 = 2°+1 ¢ 2 (2° + 1)! µ ¶ ¶2 µZ +1 x dx : g(x)p(x)H2°+1 p (36) ¢ 2¾ ¡1 If the nonlinearity corresponds to a quantization characteristic P2°+1 can be written in a closed form [14]. Since the coefficients P2°+1 are identical for the four expectations of (28), we get ³ ´ 2°+1 2°+1 2°+1 ; P2°+1 ½2°+1 + ½ + j½ ¡ j½ II QQ QI IQ
°=0
(37) with E[xI (t)xI (t ¡ ¿ )] RefRin (¿ )g = 2 E[jxI (t)j ] Rin (0) RefRin (¿ )g ¢ E[xQ (t)xQ (t ¡ ¿ )] = = ½II ½QQ = E[jxQ (t)j2 ] Rin (0) E[xQ (t)xI (t ¡ ¿ )] ImfRin (¿ )g ¢ ½QI = p = Rin (0) E[jxI (t)j2 ]E[jxQ (t)j2 ] E[xI (t)xQ (t ¡ ¿ )] ImfRin (¿ )g ¢ = p = = ¡½QI 2 2 Rin (0) E[jxI (t)j ]E[jxQ (t)j ] ¢
½II =
½IQ
+1 X °=0
2P2°+1
f2°+1 (Rin (¿ )) 2°+1
(Rin (0))
(42)
where fm (x) = (Refxg)m + j(Imfxg)m :
(43)
It should be noted that fm (x) is the sum of two terms, each one with the form of equation (9) of [14]. When the autocorrelation is real then fm (x) = xm (i.e., it has only one term) and (42) reduces to equation (9) of [14]. Acknowledgments The authors would like to acknowledge the editor, Prof. Lutz Lampe, and the anonymous reviewers for their helpful suggestions, which improved the quality of the paper. A special thanks is due to the anonymous reviewer that pointed out [14] and [15], relevant references of which we were not aware when we submitted the first version of the paper. This work was supported in part by FCT (pluriannual founding and project POSI/CPS/46701/2002 - MC-CDMA) and the C-MOBILE project IST-2005-27423. References
with p(x) denoting the probability density function of either xa or xb , i.e., ¶ µ x2 1 (35) exp ¡ 2 : p(x) = p 2¾ 2¼¾
+1 X
Rout (¿ ) =
(34)
°=0
Rout (¿ ) =
This means that the autocorrelation of the output of an I-Q nonlinearity with a bandpass, Gaussian input is given by
(38) (39) (40) (41)
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[14] D. Dardari, ”Exact Analysis of Joint Clipping and Quantization Effects in High Speed WLAN Receivers”, IEEE ICC’03, Ankorage, May 2003. [15] D. Dardari, ”Joint Clip and Quantization Effects Characterization in OFDM Receivers”, IEEE Trans. Circuits Syst. I, vol. 53, pp. 1741–1748, Aug 2006. [16] H.Rowe, ”Memoryless Nonlinearities with Gaussian Input: Elementary Results”, Bell System Tech. Journal, Vol. 61, Sep. 1982. [17] A. Bahai, M. Singh, A. Goldsmith and B. Saltzberg, ”A New Approach for Evaluating Clipping Distortion in Multicarrier Systems”, IEEE Journal on Sel. Areas in Comm., Volume 20, No. 5, June 2002. [18] D. Tufts, H. Hersey and W. Mosier, ”Effects of FFT Coefficient Quantization on Bin Frequency Response”, Proc. of the IEEE, pp. 146–147, Jan. 1972. [19] G. Szego, ”Orthogonal Polynomials”, American Mathematical Society, Colloquium Publications, Vol. No 23 4th ed., 1975. [20] M. Abramowitz and I. Stegun, ”Handbook of Mathematical Functions”, New York, Dover Publications, 1972.
