Thus Far, RF Imps, No Further!

The art of drawing the line that phase noise, intermodulation products and IQ imbalance mustn’t cross.

RF Impairments

The world isn’t ideal, why should RF circuits be any exception? If everything were perfect, mixers would mix only what we asked them to, amplifiers would stick to their linear day jobs, and oscillators would keep time better than a Swiss watchmaker. But reality prefers chaos and RF circuits are enthusiastic participants.

Many moons ago, I wrote Lost in Translation explaining how the digital signal we generate deteriorates when we convert it to analog due to the imperfections introduced by RF circuits. These imperfections are inevitable; the cost of living in the real world. All we can do is negotiate the trade-offs. How meticulously should we design our RF circuits to minimize the impairments if that means making the RF bigger and more expensive. What is our acceptable level for impairments?

That, my dear readers, is the question every systems designer gets asked by their RF counterparts. What is the noise figure you need, how good must the IIP2 and IIP3 be, what is the required phase noise profile, how much IQ imbalance can you tolerate? And you, as the systems designer, must not only provide these limits, but also justify them. How do you do that? Unfortunately, that isn’t taught in classrooms, not even written down clearly in most companies that do this kind of work, and books (who has time for them these days!) bury the big picture with equations that get you lost in the trees before seeing the forest. Fortunately for you, my dear reader, I endeavor to give you a far simpler version that immediately gets you to recognize the forest while simultaneously making the reading fun!

Let us list the four horsemen of RF impairments:

• Noise Figure: as if the ever-present thermal noise isn’t sufficient, this adds to it a bit more.
• Phase Noise: puts a skirt on each frequency making it interfere with other frequencies.
• Intermodulation: unwanted products from non-linearity can fall inband, like an uninvited guest at the party.
• IQ imbalance: creates spectral clones of frequencies and these mirror images then interfere with actual frequency components.

Let us walk through each of these to understand how to define their limits.

Noise Figure

By definition, noise figure (NF) is the number of dB by which the SNR degrades when the signal passes through an RF circuit. You start with some input SNR (because thermal noise is always there), the RF circuit then adds its own noise, and the NF tells you how much of SNR you lose at the output.

As systems designers, we compute the error vector magnitude (EVM) to specify the quality of our transmitted signal and at the receiver, the minimum RX power (sensitivity) that is required to meet the desired packet error rate.

To measure EVM, we calculate the distance between noiseless digital constellation point S = I+jQ that we wanted to transmit and the noisy analog output Sn = In+jQn. The root mean squared distance is then the EVM, which is in fact a measure of the noise addition in the analog world. Since the constellation points in digital are normalized to unit power, the EVM is just sqrt(1/SNR). That is, SNR in dB is just −20×log10(EVM). EVM is typically reported as a percentage and a 1% EVM corresponds to −20×log10(1/100) = 40 dB SNR.

Note however that this requires us to compare baseband constellation points, while what comes out of the transmitter is an RF signal modulating a carrier. We must therefore convert the RF signal to baseband before we can estimate the EVM. And make sure that this conversion from RF signal to baseband adds much less noise than what was added in the transmitter. This is one of the main reasons why the RF instruments we use in the lab are so bulky and expensive! They need high quality components to add minimal noise.

To measure the receiver sensitivity, we model the transmitter and receiver blocks, run PER simulations on an additive white Gaussian noise (AWGN) channel to find the minimum RX power, Psens, at which the desired packet error rate is met. With SNR defined as the ratio of signal power to noise power, the minimum required SNR at receiver sensitivity is Psens/Pnoise. We can then compare how close this SNR requirement is to the theoretical SNR need for the desired bit rate R, using the Shannon’s AWGN channel capacity C [bits/s] = B×log2(1+SNR).

For a system bandwidth of B Hz, the spectral efficiency is the measure of how many bits we can transmit per Hz, η=R/B. The SNR is the signal power over noise power, and noise power is it’s power spectral density N0 times the bandwidth B (because it is white!). Signal power is the total energy in each symbol divided by the symbol time Es/Ts. Since 1/Ts = B, SNR therefore is same as Es/N0. Since we are sending η bits per symbol, the energy per bit is Eb = Es/η. This gives us the following famous formula, which in the limit of η going to 0 (power limited region) gives us a minimum Eb/N0 of ln(2), which is −1.59 dB.

Even if the system itself adds no extra noise, the baseline noise floor is unavoidable because every electronic circuit at temperature T generates thermal noise. The corresponding power spectral density is N0=kT where k is the Boltzman’s constant and T is the absolute temperature in Kelvin. At room temperature (T≈290K), this evaluates to −174 dBm/Hz. The total noise power, Pnoise in dB, at the receiver input across bandwidth B is then −174 dBm/Hz + 10×log10(B) + NF (dB).

Thus, for a given data rate, if we know the SNR required for the desired packer error rate, we can compute the noise figure required to achieve the desired receiver sensitivity as:

Phase Noise

This jitter with an attitude arises from the fundamental truth that oscillators don’t oscillate perfectly. Instead of perfect sinusoid at the local oscillator frequency fLO, we get cos(2π×fLO×t + ϕ(t)), where ϕ(t) is the random process that causes jitters, literally. Because of this, the LO spectrum that ought to look like an impulse, instead wears a skirt.

The mixer performs y(t)exp(j(2π×fLO×t + ϕ(t))) and this multiplication in time becomes convolution in frequency. For ideal LO, it is convolution with an impulse function at fLO, resulting in the signal being shifted down to baseband. But since the real LO has skirts, the signal gets smeared across frequencies.

In the equivalent complex baseband model, we have every sample being rotated by exp(jϕ(t)), which is essentially the mixing operation. This means that the spectral component of ϕ(t) at some frequency f1 will translate the signal at frequency f2 to interfere with the signal frequency component at f2−f1. The smearing noise generated at f2−f1 then adds to the signal there. The power of this smeared noise depends on the power of the LO skirt at frequency fLO + f1.

This is why the phase noise skirt is specified in dBc, which is the ratio of unwanted power to the main carrier. It quantifies how much unwanted power (at offset f1​) sits relative to the carrier. If the power of the skirt is −L dBc at offset f1, the power of noise generated at f2−f1 is then Psignal(f2)−L.

If the input signal has frequencies between −B/2 to B/2, then at any given frequency f, the noise is from all signal and phase noise frequency pairs (f2, f1) such that f2−f1=f. As f1​ sweeps across the signal band, each offset f1​ has a corresponding f2=f+f1​ that contributes noise at f. The total noise at frequency f is the sum of all such contributions.

In practical terms: take the LO’s phase-noise profile, combine it with the signal bandwidth, and integrate the relevant portion. That integrated phase noise (IPN) gives you the resulting noise spectral density in the downconverted signal. The bigger the skirt, the bigger the mess.

The integration is typically from 10 KHz because offsets below that essentially give f2≈f , which is just the normal downconversion action of the LO. Everything inside that tiny neighborhood (< 10 KHz offset) is “business as usual.” We can then treat the integrated phase noise within the signal bandwidth as additive noise and use it to quantify the SNR degradation.

But this does not mean that the LO skirt outside the signal bandwidth are harmless. A phase-noise component at frequency fb​ (with power Ppn(fb)) will happily mix with any strong signal at fb​ (say, an adjacent-channel blaster with power Pb(fb)) and drag it straight into your baseband. A high-power out-of-band signal at frequency fb​ effectively appears inband at power Pb(fb)−Ppn(fb). That is how the neighbourhood 20 dBm LTE signal barges into your carefully curated baseband like it owns the spectrum.

Thus not only must we specify the integrated phase noise, but also the limits on phase noise power at different frequencies, depending on what kind of out-of-band signals are likely in our operating environment.

Intermodulation

If you push an amplifier outside its linear region, it starts coloring outside the lines. A linear amplifier sticks to y = a1×x, but a real amplifier generates y = a1×x + a2×x² + a3×x³+… If x(t) = cos(2π×f1×t) + cos(2π×f2×t), a sum of two tones, x² generates new frequencies f1+f2 and |f1−f2|, and x³ generates frequencies 2(f1)−f2 and 2(f2)−f1. There are of course other frequencies getting generated, such as the harmonics 2f1, 2f2, 3f1, 3f2, …, in addition to 2(f1)+f2, 2(f2)+f1 terms. But the moment we notice that only frequencies around the carrier fc survive the low pass filter, the dominant frqeuency terms are f1+f2, |f1-f2|, 2(f1)−f2 and 2(f2)−f1.

The magnitude of these second order frequencies are a2×A²/2, where A is the magnitude of the original tones (assuming equal power for both). We don’t know what a2 exactly is, but we see that for every dB increase in the input signal power, the second order components increase by 2 dB. The input-output relationship is therefore linear in the log domain!

Similarly, the magnitude for third order components is 3×a3×A³/4. For every dB increase in input, the third order component increases by 3 dB. Thus, the slope for 2nd order intermodulation components is 2 and the slope for 3rd order intermodulation components is 3.

So we know the rate at which the magnitude of the 2nd and 3rd order intermodulation components increase, but what is their magnitude relative to the first order component? To find that, we have two lines y = x (linear component) and y = 2x−c (2nd order IMD). When these second line intersects the first line at some x value, we have x = c, and this is the 2nd order input intercept point, IIP2. Similarly, the 3rd order input intercept point IIP3 is obtained frm y = 3x -c3.

Once we know IIP2/IIP3, we can determine the power of the 2nd and 3rd order components using the dBm power of the input, and we get P_IMD2 = 2Pin−IIP2 and P_IMD3 = 3Pin−IIP3.

An input signal of bandwidth B already contains multiple frequencies between −B/2 to B/2. The intermodulation components created by these frequencies show up as additive noise with power P_IMD. As long as this P_IMD is well below the thermal noise floor, it does not cause additional harm. As an example, for an IIP2 of 30 dBm and Pin of -60 dBm, we have P_IMD2 = 2×(−60)−30 = −150 dBm. Since this is well below the noise floor, the IMD noise here does not limit the SNR. For an IIP3 of -10 dBm, we have P_IMD3 = 3×(−60)−(−10)= −170 dBm, which again is way below the thermal noise floor.

Does this mean the intermodulation noise is not that critical? They absolutely are! To see why, consider a strong out-of-band blocker with two tones f1 and f2. If |f1-f2| is less than our bandwidth of interest, the IMD2 noise then falls inband. For a -20 dBm out-of-band blocker, we get a P_IMD2 of 2×(−20)−30 = −70 dBm. This scenario could easily happen if the same device transmits simultaneously on another frequency: say WLAN transmitting on 2.4 GHz and receiving on 5 GHz. The +20 dBm transmission at 2.4 GHz could easily be picked up by the 5 GHz receiver front end to generate IMD noise. Any passive bandpass filters at 5 GHz front end will only have ~40 dB attenuation and we would still get −20 dBm signal at the receiver’s RF input.

The IIP2 and IIP3 specs are thus more critical for performance in presence of blocking signals, even if those blockers are on a far off frequency. The IIP3 spec is important for the low noise amplifier (LNA) at the receiver’s RF front end, since IM2 components |f1−f2| usually ends up at low frequency or high frequency instead of at the carrier frequency fc. For the mixer though, the IIP2 spec is important since the low frequency |f1−f2| component becomes the common mode for the differential circuit and a finite common mode rejection results in IMD2 noise getting added to baseband.

IQ Imbalance

In an ideal quadrature receiver, I and Q paths have equal gain and their phases differ by exactly 90 degree. In real life, the LO for I and Q path are rarely perfectly aligned. The unequal gain between cos and sin creates gain imbalance, while additional phase offsets introduce phase imbalance. The equations below show how this results in an image being generated at −f for every signal frequency f.

This image creates unwanted frequency components that interfere with signal frequency components at −f. We quantify this noise via the image rejection ratio (IMRR):

Higher the IMRR, lower the unwanted image frequencies compared to signal frequencies. IMRR is therefore a measure of effective SNR when the noise orignates from IQ imbalance. Unlike the other impairments, IQ imbalance can be compensated digitally via calibration and this can improve IMRR without changin the RF circuits.

The Cumulative Effect

Every impairment we have discussed so far effectively adds noise.

• Noise figure directly degrades the SNR.
• Phase noise degrades SNR via the integrated phase noise.
• IMD2/IMD3 creates intermodulation frequencies that interfere will signal frequencies and adds to the noise floor
• IMRR due to IQ imbalance limits the SNR due to the image component adding to the noise floor

We have multiple noise sources here and we need to tie them together to find out how much the SNR degrades. For noise figure, the SNR degradation is easy to see: SNR_NF [dB] = SNR_in [dB]−NF [dB].

The integrated phase noise is a dBc value indicating the noise level relative to the input signal power: IPN[dBc]= P_ipn[dBm]−P_in[dBm]. That is, P_in/P_ipn = 10^(−IPN[dBc]/10). SNR is P_in/P_noise, and since the noise terms P_noise, P_ipn are to be added, we need to do this in the inverse SNR domain.

The IMD noise power P_imd [dBm] is an absolute value and we can compute a Δimd[dB] = P_imd [dBm]−P_noise [dBm]. The inverse SNR can then be adjusted as:

Finally, the IMRR is P_in/P_image, and we therefore have the combined inverse SNR as:

This final expression is the RF-equivalent of adding up everyone’s contribution to a group project — except instead of work, everyone contributes noise. The receiver’s effective SNR is governed by the sum of:

• thermal noise (via NF),
• phase-noise skirt leakage,
• IMD splash-over,
• and IQ image leakage.

Everything is added as equivalent noise power, and only then inverted to retrieve the final SNR. If one term dominates, the others barely matter; if several are comparable, they gang up like villains in a multiverse crossover event.

You now have the full toolkit: how noise figure sets the baseline, how phase noise smears the spectrum, how intermodulation creates spectral graffiti, and how IQ imbalance spawns spectral doppelgängers that nobody invited. Each of these misbehaviors has a clean, quantifiable way of degrading SNR, and, more importantly, a clean way for you to specify how much is too much. This is the whole art of RF-system design in one sentence: decide how much imperfection the system can tolerate, then politely but firmly tell the RF circuitry the limits it shall not exceed.

And that, my dear readers, is how you draw the line that the RF imps must not cross!