In order to gain insight into the impact of typical impairments, not only in terms of the impact on a single-user’s communication link but also in terms of their overall effect on system-level spectral efficiency, it is instructive to develop a more precise analytical model. In particular, three sources of errors are included here:
• Phase noise;
• Modulator image interference (amplitude and phase imbalance);
• Non-linear distortion and intermodulation products which fall into adjacent RBs.
27Additive White Gaussian Noise. Note that the reasoning can be straightforwardly extended to the case of a frequency-selective convolutive channel and an OFDM system with CP.
28Coherent detection means that the phase of the received signal is known to the receiver a priori, for example from Reference Signals (RSs) as discussed in Chapter 8.
The model here is developed considering an OFDM signal (downlink) but it can be easily generalized for an SC-FDMA signal (uplink). The OFDM signal can be written as the sum of closely-spaced tones which do not interfere with each other due to their orthogonality. In particular the discrete-time signal for symbolis written as
x[n]=
N/2−1 k=−N/2
Sk,exp
j2πkΔf n N
(21.8) where N is the number of subcarriers, Sk, is the constellation symbol sent on the kth subcarrier for thethOFDM symbol andnrepresents the component within theth OFDM symbol,n∈[0,N−1]. In the following, we can drop the dependency from the variable without loss of generality.
21.5.2.1 Amplitude and Phase Imbalance
Let us consider the continuous time version of the signalx(t) modulated to frequency fc, i.e.
r(t)=Re{x(t)ej2πfct}=x(t)ej2πfct+x(t)e−j2πfct (21.9) With a non-ideal I/Q (de-)modulator, the recovered signal can be written as follows:
yIQ(t)=LP{r(t)(cos(2πfct)−jβsin(2πfct+φ))}
(1)=LP{(x(t)ej2πfct+x(t)e−j2πfct)(γ1e−j2πfct+γ2ej2πfct)}
(2)=x(t)γ1+x(t)γ2 (21.10)
where LP means Low-Pass filtering, (1) is because of Equation (21.9) with γ1=(1+ βe−jφ)/2 and γ2=(1−βe−jφ)/2 and (2) is obtained by low-pass filtering the signal. By substituting Equation (21.8) into (21.10), after a variable change it follows that
yIQ(t)=
N/2−1
k=−N/2
1+βe−jφ
2 Sk,+1−βe−jφ 2 S−k,
exp
j2πkΔf t N
(21.11) By lettingβ=1+qandQ(β, φ)=(q−jφ−q jφ)/2, using the approximation that cos(φ)=1 and sin(φ)=φfor small angleφ, it follows that
yIQ(t)=
N/2−1 k=−N/2
Sk,exp
j2πkΔf t N
+Q(β, φ)
N/2−1
k=−N/2
(Sk,−S−k, ) exp
j2πkΔf t N
(21.12) Equation (21.12) shows clearly that the I/Q imbalance creates two different error terms: the first is the self-interference created by the same signal at the same subcarrier frequency, while the second is the signal at the frequency mirror-image subcarrier. The ideal case corresponds
toq=0 andφ=0. Note also that the amplitude imbalance and the phase imbalance create the same effect, i.e. only the factorQ(β, φ) changes, in case of only amplitude imbalance being present (φ=0) Q(β, φ)=q/2, and when only phase imbalance is present the multiplicative coefficient becomesQ(β, φ)=jφ/2.
For high-order modulation (increasing number of states in the constellation) the error term acts as noise and it spreads the constellation points as shown in [15].
In general the amplitude and phase imbalance can vary depending on the frequency, on a subcarrier basis. This happens when there is a timing offset between the in-phase and quadrature signal paths.
21.5.2.2 Phase Noise
When the LO frequency at the transmitter is not matched to the LO at the receiver, the frequency difference implies a shift of the received signal spectrum at the baseband. In OFDM, this creates a misalignment between the bins of the FFT and the peaks of the sinc pulses of the received signal. This results in a loss of orthogonality between the subcarriers and a leakage between them. Each subcarrier interferes with every other (although the effect is strongest on adjacent subcarriers) and, as there are many subcarriers, this is a random process equivalent to Gaussian noise. Thus, a LO frequency offset lowers the SINR of the receiver. An LTE receiver will need to track and compensate for this LO offset and quickly reduce it to substantially less than the 15 kHz subcarrier spacing. This is a tougher requirement than for UMTS. The phase noise impairment has been widely studied both for a single antenna [15–17] and for the MIMO case [18], especially in the context of WiMAX.
The ideal baseband transmitted signal, neglecting the additive white Gaussian noise, is given in Equation (21.8). The received baseband signal, in the presence of only phase noise can be written as
yθ(t)=
N/2−1
k=−N/2
Sk,exp
j2πkΔf t N
ejθ(t) (21.13)
whereθis the phase noise. In [11] it is shown that the single-sideband phase noise power follows a Lorentzian spectrum [11, 19]:
L(f)= 2 πΔf3dB
1
1+[2f/(Δf3dB)]2 (21.14) whereΔf3dBis the two-sided 3 dB bandwidth of phase noise. The power spectrum given in Equation (21.14) can be considered as an approximation to practical oscillator spectra.
After the FFT, the symbol transmitted on themthsubcarrier in thethOFDM symbol can be written as
zm,=Sm,1 N
N/2−1
n=−N/2
ejθ(n)+ 1 N
N/2−1
k=−N/2,km
Sk,
N/2−1
n=−N/2
exp
j2πΔf n N(k−m)
ejθ(n) (21.15) By definingC(k)=(1/N)(N/2−1
n=−N/2ej2πΔf nk/N+jθ(n), as in [16], Equation (21.15) can be rewrit- ten as
zm,=Sm,C(0)+
N/2−1 k=−N/2,km
Sk,C(k−m) (21.16)
Equation (21.16) shows that the effect of phase noise is twofold: the useful symbol transmitted on themthsubcarrier is scaled by a coefficientC(0) which depends on the phase noise realization on the th OFDM symbol, but it is independent of the subcarrier index – i.e. all the subcarriers are rotated by the same quantity C(0). This is referred to as the Common Phase Rotation (CPR). This term can be estimated from the RSs and removed. In LTE, four symbols per subframe contain RS (see Section 8.2), so that in theory low frequency phase noise up to about 2 kHz can be compensated. However common loop bandwidths of PLLs29are in general in the order of 10 to 100 kHz, so that a major part of the phase noise energy is outside the region which can be compensated. The second term causes Inter-Carrier Interference (ICI), the amount of which is given by the coefficientC(k−m) for subcarrierk interfering on subcarrierm. The ICI due to phase noise creates a fuzzy constellation [15].
In [18] the above equations are generalized for the MIMO case, showing that phase noise has similar effects in that case.
Time-domain offsets
Any timing synchronization error results in a misalignment between the samples contained in the OFDM symbol and the samples actually processed by the FFT. If the timing error is large, some samples could be from the wrong symbol, which would cause a serious error. It is more likely that the timing error will be just a few samples and the presence of the cyclic prefix should give enough margin to prevent samples from the wrong symbol being used. Assuming this is the case, the shift in time is equivalent to a linearly-increasing phase rotation of the constellation points.
The use of higher modulation schemes and wider channel bandwidths mean that timing synchronization needs to be performed more accurately for LTE than for UMTS.
Another source of timing error is a sampling frequency error. This can occur when the baseband sampling rate of the receiver is offset from that of the transmitter, or if there is jitter of the receiver’s clock. When the sampling rate is too high this contracts the spectrum of the signal; similarly, slow sampling widens the spectrum. Either of these effects misaligns the FFT bin locations, resulting in a loss of orthogonality and hence reduced SINR. A sampling rate converter, perhaps driven by an error estimate, could correct for a systematic sampling rate error or sampling rate drift. Sampling jitter should be kept low by the choice of a clean crystal reference oscillator.
Group delay distortion
Filter group delay and amplitude ripple variation create deviation from the wanted impulse response of the receiver and cause inter-symbol interference. Unfortunately analogue filters cannot have both flat group delay and flat amplitude response, so a compromise is needed.
To achieve this the analogue channel filter is usually slightly larger than the channel to allow reduced in-band distortion at the expense of out-of-band attenuation and in-band noise suppression.
One of the inherent advantages of multicarrier OFDM is that it is quite resilient to group delay variation and amplitude ripple, more so than the single-carrier QPSK modulation used by UMTS.
29Phase-Locked Loop.
Additionally, the OFDM symbol rate is relatively low and thus there are greater margins for delays introduced by large filters. Overall there is more freedom with LTE to design the receiver analogue and digital filters to achieve high selectivity.
Reciprocal mixing
An unwanted phase offset of the received signal may include a large fixed (or slowly varying), phase offset, which can be corrected by the equalizer. However, the LO also introduces small random variations and jitter of the frequency and phase, which manifests itself as shaped phase noise, tending to reduce with offset from the carrier frequency. This cannot be corrected by the equalizer. Crystal oscillators have low levels of phase noise, but synthesizers are not so clean, especially if they are PLL-based.
Each of the elements of a frequency synthesizer produces noise which contributes to the overall noise appearing at the output. The noise within the PLL loop bandwidth arises from the phase detector and the reference, whereas outside the loop bandwidth the VCO30 is the main noise source. The phase noise tends to reduce from the edge of the PLL loop bandwidth until eventually it reaches a flat noise floor.
A serious problem for receivers caused by phase noise is reciprocal mixing. Reciprocal mixing occurs when the phase noise on the receiver LO mixes with a strong interfering signal to produce a signal which falls inside the pass-band of the receiver. Intermediate-Frequency (IF) filters, if used, do not give any rejection of such signals; instead the problem must be solved by keeping the LO phase noise at a sufficiently low level. The various interference requirements (see Sections 21.4.6 to 21.4.7.2) together create an overall requirement for the receiver LO phase noise. The interferer which gets mixed onto the received signal needs to be weaker by a margin of the SINR requirement plus the implementation margin, plus a further margin of about 10 dB if it is to have no impact on the received signal. Therefore, the LO noise,L, needs to satisfy the following requirement:
L≤SINR+IM−C/I+10+10ãlog10(B) (dBc/Hz)
21.5.2.3 Non-Linear Distortion and Intermodulation Products
IMD results from non-linearities of the PAs. The high PAPR (see Section 5.2.2) associated with multicarrier signals is one of the principal challenges in the implementation of OFDM systems. It requires linear operation of the PA over a large dynamic range, and this imposes a considerable implementation cost and reduced efficiency. The linearity and efficiency of a power amplifier are mutually exclusive specifications.
As already discussed, practical power amplifiers have a non-linear response. Numerous models exist, a selection of which can be found in [20]. Here we describe a simple polynomial memoryless model, by which the output signal can be written as
yPA(t)=a1y(t)+a2y2(t)+a3y3(t) (21.17)
30Voltage-Controlled Oscillator.
wherea1, a2, a3 are independent coefficients which can be found by measurement. From Equation (21.8), (21.17) can be written as31
yPA(t)=a1
N/2−1
k=−N/2
Sk,exp
j2πkΔf t N
+a2
N/2−1 k=−N/2
N/2−1
p=−N/2
Sk,Sp,exp
j2π(k+p)Δf t N
+a3
N/2−1 k=−N/2
N/2−1
p=−N/2 N/2−1
v=−N/2
Sk,Sp,Sv,exp
j2π(k+p+v)Δf t N
(21.18) The non-linear response of the PA creates ICI. The intermodulation products contribute to a noise-like cloud surrounding each constellation point.32 For higher-order modulation in particular (such as 64QAM), these constellation clouds contribute to an increase in error rate for each subcarrier. Thus, in an OFDM modem design, linearity must be carefully controlled.