Lost in Translation: RF Impairments and Signal Distortion

A look at the various effects that affect the digital signal when it is converted to analog for propagation via radio frequency.

We live in a reality where time flows continuously and therefore the waveform that we transmit are also continous-time. Though we may generate the data to be transmitted digitally, we must convert that from the discrete-time to continous-time analog signals for them to be transmitted. While we generate a certain range of frequencies from 0 to B Hz (if the signal is real, then its spectrum occupies -B to B Hz), this baseband signal cannot be transmitted over air. Because for an antenna to radiate the electromagnetic waves, its dimensions must be on the order of the wavelength λ of the signal. For a dipole antenna, the length of the radiating element is λ/2 for it to resonate and create standing waves. Baseband signals have a large wavelength and would require large antenna dimensions. Radio frequency (RF) signals have wavelenghts that are lower than 30 cm and are better suited for wireless transmission. Therefore, we need to upconvert the signal to RF frequency at the transmitter, and then downconvert from RF to baseband (or low IF — intermediate frequency) at the receiver.

Electromagnetic (EM) radiation occurs due to a changing electric field inducing a changing magnetic field, which in turn induces a changing electric field. This self-sustaining process propagates as an electromagnetic wave. In transmission lines, electromagnetic radiation is minimized by ensuring that the magnetic fields generated by opposing currents cancel out, confining the energy within the transmission medium. This is achieved by inducing an electric current in the opposite direction by having two conductors separated by a dielectric. Take the coaxial cable for example, the inner wire and the outer conductor are separated by a thin insulator, and when current is generated in the inner wire in one direction, an equal current is induced in the outer conductor in the oppostite direction. This results in two magnetic fields around the coaxial cable that cancell each other out. Similar principle is in action on the microstrip lines of the PCB. These strip lines are separated from a ground plane by a thin dielectric, and when current travels through these strip lines, an equivalent current is induced in the ground place in the opposite direction.

If we generate the baseband signal s(t) directly in analog form, it is called analog modulation. The signal s(t) can be used to vary the amplitude of an RF carrier (resulting in amplitude modulation, AM) or the frequency of the RF carrier (resulting in frequency modulation, FM). Changing the frequency is equivalent to changing the phase over time, since frequency is the time derivative of phase: f = dϕ/dt. If we generate the discrete-time sequence s[n], we call this digital modulation. In this case, the information is typically mapped to discrete amplitude, phase, or frequency levels before being upconverted to an RF carrier.

Digitally generated discrete signals have periodic spectrum. Passing this through a digital-to-analog convertor (DAC) retains the periodic spectral components, known as DAC images or spectral replicas. An analog low pass filter (reconstruction filter) is required to remove these unwanted images and recover the desired continous-time signal. A zero order hold DAC, where each sample is held for one sampling period, results in the DAC images being shaped by the sinc response. But this is usually not enough suppression since the transmit signal needs to meet stict out-of-band emission limits. Depeding on the additional image rejection required, a second order Butterworth or Chebychev analog low pass filter is applied in practice.

When we transmit QAM symbols, s[n]=∑c[k]p[n−kL] where the summation is over k, p[n] is the pulse shape, and L is the oversampling factor. The transmitted signal is generated by upsampling and pulse shaping the QAM symbols. In OFDM systems, the larger OFDM symbol is generated via FFT, CP is added and pulse shaping is applied to the OFDM symbol. Following image shows the pulse shaping of 16-QAM symbols and its corresponding spectrum. These sequence of discrete samples s[n] = sI[n] + jsQ[n] are converted to the analog baseband signal s(t) = sI(t) + jsQ(t). This means we have one DAC+LPF for the I path and another DAC+LPF for the Q path.

If the pulse shape p[n] is chosen in such a way that its impulse response is not limited to one symbol perid, we then generate intersymbol interference (ISI). We can avoid ISI on the transmitter, but as the signal passes through analog filters and over a multipath channel, the resuling pulse spread inevitably causes ISI. The eye diagram, which is nothing but overlapping of the pulses over an integer multiple of the symbol period, shows how severe the ISI is. The following image shows the eye diagram of the real part of pulse shaped QAM (over two symbol periods) and the eye diagram after the signal passing through an analog Chebychev LPF.

We can see that there is already some distortion introduced when the signals pass through this analog LPF. This is due to the non-linear phase response of the analog LPF which results in a frequency dependent group delay — different frequency components are delayed differently and combined at the filter output. The effect of this group delay distortion can be seen in the constellation plot (image below), which is generated by sampling each symbol at the peak of its pulse.

Since the I and Q axes on the complex plane are perpendicular to each other, we use cos(ωt) for I path and sin(ωt) for Q path to upconvert the baseband signal to RF. The transmitted RF signal is then x(t) = sI(t) cos(ωt) + sQ(t) sin(ωt). The transmitter and receiver block diagram is shown in the following image.

At the receiver, the I path is recovered by multiplying the received signal with cos(ωt) and only the baseband signal is retained by applying a low pass filter (LPF). Similarly, the Q path is recovered by mixing with sin(ωt) followed by LPF. Applying the low pass filter removes all frequencies that are above the signal bandiwdth B. The equations are written out in the image below.

The trignometric identities used in these equations can be derived from sin(A+B) = sin(A)cos(B)+cos(A)sin(B) and cos(A+B)=cos(A)cos(B)-sin(A)sin(B). The above set of equations can be simplified if we use the complex exponential exp(jωt)=cos(ωt)+j sin(ωt). The resulting expressions for the RF transmit signal x(t) at the TX antenna and received baseband signal y(t) are shown in the image below. We have made use the fact that for complex number z, real part is (z + *z)/2 where *z* is the complex conjugate of z and imaginary part is (z - z)/2j.

What we have ignored in these equations is the presence of additive noise. Every analog circuit adds thermal noise to the signal that is the result of random motion of electrons at the given temparature. The power in watts of this noise is kTB where k is the Boltzmann’s constant (that relates the average kinetic energy of particles to the temperature of the system), T is the temperature in Kelvin and B is the bandwidth of the system in Hz. For B=1 Hz and room temperature (300K = 27 degree celcius), the thermal noise power is -174 dBm/Hz.

For a WiFi system with 20 MHz bandwidth, the thermal noise level will already be -174 dBm/Hz + 10 log10(20e⁶) = -101 dBm. Any additional noise that is added by the analog circuit is called its noise figure. It is the difference of input SNR and output SNR in dB. If the low noise amplifier (LNA) adds 3 dB additional noise, its noise figure is 3 dB. Any amplifier not just amplifies the input signal, but also the input noise, but if no additional noise is added, then the input and output SNR must remain the same.

If a receiver requires a certain SNR to reliably recover the QAM symbols, then the sensitivity is noise level + SNR. The noise level includes the noise figure of the receiver. For the WiFi system example, if we need 3 dB SNR for the lowest modulation, BPSK, and if the analog receiver has 4 dB noise figure, then the sensitivity for the lowest rate is -101 dBm + 4 dB + 3 dB = -94 dBm. The model for an amplifier with gain G then is y = Gx + n, where the output y is an amplified version of input x and there is noise addition on top. The noise factor F is then SNR(y)/SNR(x).

If there are multiple gain stages, each with gains G1, G2, etc., and the corresponding noise factors F1, F2, etc., what is the total noise factor? See that every subsequent stage adds noise to an input that is already amplified by the gain of the previous stages, so any additional noise contribution must be scaled down by the gains of all previous stages, since the total noise factor F is ratio of the SNR of output of the last gain stage with the SNR of the input of the first gain stage. This gives us the Friis equation for computing the total noise factor:

As a result, the noise figure = 10 log10(nosie factor) of the first stage dominates as long as the first stage has a high gain. This is why the first stage of the receiver is the “low noise” amplifier that typically has 15 to 20 dB gain and a 3 to 4 dB noise figure.

This thermal noise is seen to follow Gaussian distribution and its power spectral density (Fourier transform of the autocorrelation) is uniform. That is, every noise sample is independent and identically distributed (iid), and the iid Gaussian noise is therefore white and is additive in our model, hence AWGN (additive white Gaussian noise). The following image shows the 16-QAM constellation plot with AWGN noise for different SNR values.

Electronic circuits also have what is called flicker noise, or 1/f noise since its power spectral density is inversely proportional to some exponent of f. For the local oscillator (LO) that generates the carrier frequency, this noise causes random phase changes over time. We can model this as the oscillator generating exp(j(ωt + ϕ(t))) where ϕ(t) is the phase noise. The power spectral density of ϕ(t) as seen in practical systems is empirically modelled by Leeson’s equations:

In the above equation, F is the thermal noise factor of the oscillator circuit, k is the Boltzmann constant, T is the absolute temperature, Ps is the output power, fLO is the local oscillator output frequency, Q is the quality factor of the oscillator, fα is the corner frequency for 1/f noise and f is the frequency offset from fLO. For f << fα, Sϕ(f) is dominated by 1/f³ term and gives a slope of -30 dB/decade. For fα < f < fb=fLO/2Q (the natural bandwidth of the oscillator), Sϕ(f) is dominated by 1/f² term and therefore has a slope of -20 dB/decade. For f >> fb, thermal noise dominates. Image below shows a typical power spectral density of phase noise and its effect on the carrier.

To generate ϕ(t) from Sϕ(f), we first note that ϕ(t) is the integral from 0 to t of the frequency noise Δf(t) that causes instantaneous fluctuations in frequency. The power spectral densities Sϕ(f) and SΔf(f) are related by Sϕ(f) =SΔf(f)/(2πf)² which is a direct result of the Fourier transform property where differentiation and integration in the time domain correspond to multiplication and division by j2πf in the frequency domain respectively. We can now generate a white noise process W(f) in frequency, shape it with the power spectral density of frequency noise SΔf(f) that can derived from the power spectral density of phase noise Sϕ(f), transform this to time domain noise by applying IFFT, and computing the cumulative intergal/summation of this frequency noise samples Δf(t) to get phase noise samples ϕ(t).

Assuming a relative phase offset at the receiver compared to the transmitter, we can compute y(t) = LPF{x(t)exp(j(ωt + ϕ(t)))} and this will give us y(t) = s(t)exp(jϕ(t)). We can now have ϕ(t) as the phase noise component, or as the phase due to frequency offset ωo, ϕ(t)=(ωo)t. The corresponding effect on the constellation is shown in the plot below.

In addition to the phase offset in the carrier between the transmitter and receiver, there could be phase offsets between the cos(ωt) and sin(ωt) too! We could model this as relative phase offset only on sin(ωt+ϕ). To see the impact of this on the baseband signal y(t), we can use the identities cos(θ) = (exp(jθ)+exp(-jθ))/2 and sin(θ) = (exp(jθ)-exp(-jθ))/(2j). The derivation is shown in the image below.

If we instead use sin(ωt+ϕ), we can see that we get mixing of the real and imaginary parts of s(t).

We could also have gain imbalance between cos and sin, in addition to the phase imbalance . We can write the complex exponential then with the gain and phase imbalance and derive y(t) as shown in the image below.

Taking the Fourier transform, we have Y(f) = S(f)(1+g exp(jϕ)) + S(-f)(1-g exp(jϕ)). In the ideal case, when g=1 and ϕ=0, we have Y(f) = S(f). In the non-ideal case, there is an additional image created and the ratio of the wanted signal magnitude to the image magnitude is the image rejection ratio: IRR = (1+g exp(jϕ))/(1-g exp(jϕ))*. If we send a complex tone whose ideal baseband spectrum will have a peak at the tone frequency, in a system with IQ imbalance, there will also be a peak at the negative tone frequency (the image).

At baseband, this combining of I and Q can be modelled as a matrix multiplication, and the compensation is then simply the inverse of this matrix. The effect of this IQ imbalance on the constellation is shown in the image below. We see that constallation is skewed by the gain and phase imbalances.

In our analog model, we have only multiplications and additions which are linear operations. But the signals we send have high peak to average power ratio (PAPR), unless we use constant amplitude modulation like PSK. OFDM systems have about 10 dB PAPR for higher QAM. This requires all the RF circuits to be linear over a wide range of inputs. As with everything else in RF, this is never the case and we have non-linear behavior at certain range of input power levels. For an input x, the non-linearity can be modelled as y = ax + bx² + cx³+… and if x is a single tone at frequency f, the x² and x³ terms will generate tones at 2f and 3f.

The second and third order non-linearities are usually the critical ones. Typically, mixer generates second order non-linearities and the amplifiers generate third order non-linearities. For amplifiers operating at RF frequencies, the second order and third order harmonics will be far away from our signal of interest, but when we have multiple frequency tones, these frequencies mix together to generate intermodulation distortion (IMD). For example, if there are two tones with frequencies f1 and f2, the second order non-linearity will generate f1+f2 and f1-f2 in addition to 2f1 and 2f2, while the third order non-linearity will generate 2f1-f2, 2f2-f1, 2f1+f2, 2f2+2f1. The frequencies 2f1-f2, 2f2-f1 could fall inband.

We quantify these non-linearities by looking at the power level where the tone power and the power of the second or third order product become the same. For the second order product, this power is called the second order intercept point (IP2) and similarly, we have third order intercept point (IP3). The following image taken from web shows why this is called an intercept point and how it is calculated.

As we increase the input tone power, the output tone power increases and so does the power of the IMD products. At some input power, these output tone power and the power of IMD product will become the same and that power is our intercept point. The P1db point is that output power level which is 1dB below where it should be if the amplifier was linear. The effect of non-linearity on the constellation is also shown in the image.

These distortions impact the error vector magnitude (EVM) which is a critical metric to measure the quality of the transmitted signal. To estimate EVM, the received signal must be mapped to its most likely constellation point, and the absolute squared error between the received symbol yk and the estimated constellation point sk is calculated. That is, EVM(%) = sqrt(∑|sk-yk|²)/sqrt(∑|sk|²) × 100. If we have 1% EVM, it is equivalent to 20 log10 (0.01) = -40 dB. Because EVM is a magnitude ratio, dB conversion for magnitude (not power) uses 20 log10.

Higher EVM(%) reduces the effective Signal-to-Noise Ratio (SNR) at the receiver, degrading system performance. To maintain acceptable EVM levels, transmit power must be adjusted based on the Modulation and Coding Scheme (MCS). Higher-order modulation schemes, such as 64-QAM or 256-QAM, require lower EVM(%) and this requires lowering the transmission power compared to keep the power amplifier (PA) in linear region. Power amplifiers exhibit both amplitude and phase distortions. AM-AM distortion describes the non-linear gain compression, while AM-PM distortion refers to the phase shift introduced as a function of input power. To characterize these non-linearities, polynomial models are commonly used.

If we can characterize the PA response, either as via the AM-AM and AM-PM curves or by curve fitting to a polynomial model, we can invert this non-linear mapping digitally by predistorting the input signal such that when the distortion due to PA non-linearity is applied, the effective result is a linear response. This is called digital pre-distortion (DPD). If we have a memoryless PA, meaning the output only depends on the current input, then the PA characteristic can be broken down in a piecewise linear model and at every power level, we then have a complex scale factor that can then be inverted by DPD.

But for wideband signals, PA starts exhibiting memory effects, meaning its output depends not only on the current input but also on past inputs. This occurs due to thermal effects, charge storage in active devices, and frequency-dependent gain variations. Memory effects can be modeled as: ∑∑h(m,n)x(t-m)^n, where the summartions are over m and n. DPD in this case is more complex, requiring iterative methods or machine learning techniques.

Before we wrap up, let us just cover one more interesting topic. The power amplifiers has one of the highest power consumption in the wireless system. This is because the PA in its linear region has relatively low efficiency (< 30%). Efficiency is a measure of how much of the supplied power is converted into output power. If PA output is close to its supply voltage, we are are in the non-linear region of the amplifier where the efficiency is ~ 70%. In linear region, the supply voltage is much higher than the output power, resuling in the supply voltage dissipating as heat instead of being converted to RF power.

If we can reduce the PA supply voltage when input signal level is low, we can always operate the PA with high efficiency, thereby saving power consumption. We can do this by changing the PA supply voltage based on the input signal envelope, and this technique is called envelope tracking (ET). However, if we do this, we would be always operating the PA in the non-linear region and therefore would require DPD to compensate for this non-linearity. In this case, DPD not only needs to learn the non-linearity for each input signal level, but also needs to learn how this non-linearity changes as the PA supply voltage changes.

How do we learn the PA characteristics and compute the inverse function for DPD? How do we estimate the IQ imbalance we have in our transmitter and receiver? We still have other impairments, such as the local oscillator feed through (LOFT) at the transmitter, the DC offsets at the receiver, spurs that couple from RF PLL and baseband PLL and in case of parallel ADCs, the gain imbalance and timing offsets per ADC. Calibrating out these impairments becomes critical for a high performance transceiver. And this is usually where most of the secret sauce is between different chip vendors. How cleverly can we design the self-calibration algorithms so that we compensate as much of these impairments as possible while minimizing the expensive device calibration or factory calibration requiring external instruments. Perhaps there is another article to be written on the estimation and calibration of these impairments. I hope this one still provides some insights into all the trouble we get into when we convert from digital to analog and back.