Mixing, Noise, and Signal-to-Noise Ratio in Diode Mixers

Introduction

Mixers are one of the most important building blocks in RF systems. They allow signals to be shifted in frequency - from RF down to an intermediate frequency (IF), or all the way to baseband. This enables filtering, amplification, and detection in more convenient frequency ranges.

However, mixers don’t just act on the desired signal. They also transform noise, and the way noise behaves is often counter-intuitive. In this article we’ll look at how diode mixers work, how signal and noise behave under mixing, why image folding can degrade the signal to noise ratio (SNR), and how local oscillator (LO) noise plays a role. Finally, we’ll show some measurements that illustrate these effects in practice.

The following article tries to take a look at some basic concepts - the SNR ratio and the signal loss over mixers - in an intuitive way.

• How Diode Mixers Work
• Signal and Noise
  • Signal
  • Noise
• Image Folding
  • Case 1: Image rejected
  • Case 2: Both images pass (folding)
• Phase Noise and LO Noise
• Homodyne vs Heterodyne (Recap)
• Measurements
  • The (near) homodyne configuration
  • The heterodyne configuration
  • Comparison
• Possible countermeasures
• Conclusion

How Diode Mixers Work

Mathematically, an ideal mixer multiplies the input signal with the local oscillator LO:

[ y(t) = x(t) \cdot \cos(\omega_\text{LO} t) ]

Here $x(t)$ is the input signal and $\omega_\text{LO} = 2 \pi f_\text{LO}$ the (angular) frequency of the local oscillator. If the RF input is a single tone $x(t)$ at the frequency $\omega_\text{RF} = 2 \pi f_\text{RF}$

[ x(t) = A_s \cos(\omega_\text{RF} t + \phi_s) ]

the output is given by the multiplication

[ y(t) = \frac{A_s}{2} \left( \cos((\omega_\text{RF}+\omega_\text{LO})t) + \cos((\omega_\text{RF}-\omega_\text{LO})t) \right) ]

As one can see the mixer yields two signals - one at the sum of the frequencies of the RF signal and the LO, one at the difference frequency. Filtering after the mixer optionally selects the one band we want.

In practice, double-balanced diode mixers use a ring of diodes driven by a strong LO (e.g. +7 dBm) so they fully switch into conductive mode even for weak signals on the RF input. The LO switches the diodes rapidly, chopping the RF input and the so called IF port collects the resulting sum and difference products. This switching operation makes diode mixers passive (no gain) and gives them a conversion loss of typically 6-8 dB. Note that you can also use diod mixers in reverse configuration for modulating a signal - you input a low frequency signal at the IF port, apply the local oscillator to the LO port and get the mixed output at the RF port.

Signal and Noise

Signal

For a single tone (i.e. a single frequency $\omega_\text{RF} = 2 \pi f_\text{RF}$) the multiplication according to the rules for sine and cosine multiplication splits the signal into sum and difference tones. If we select only one of them the power is reduced by 3 dB (a factor of $\frac{1}{2}$). In addition a real passive mixer adds another ~6–8 dB conversion loss.

So the IF signal power is

[ P_\text{IF,signal} \approx P_\text{RF,signal} - L_c, ]

with $L_c$ the conversion loss.

Noise

Noise is different: it is not a single tone, but - assuming white noise - a flat spectrum of random tones. Its spectral density at room temperature is for example

[ N_0 \approx -174 \,\text{dBm/Hz} ]

and in a limited bandwidth $B$ (for example by filtering),

[ P_\text{noise} = N_0 + 10\log_{10} B ]

When noise passes through a mixer the spectrum is duplicated and shifted

[ Y(f) = \tfrac{1}{2}\left(X(f-f_\text{LO}) + X(f+f_\text{LO})\right) ]

That means both sidebands around the LO carry identical copies of the noise spectrum. Whether this hurts SNR depends on whether both copies end up overlapping or not (which is called image folding).

Image Folding

Case 1: Image rejected

If your IF filter keeps only one sideband which is typical when applying bandpass filtering in the intermediate frequency section (usually the lower difference band), then the noise spectral density in your measurement bandwidth is unchanged. You get the same noise power as before in bandwidth $B$. Signal and noise both survive proportionally.

This means that SNR is preserved (apart from conversion loss). Intuitively one would assume to get half of the noise - this is not true. Noise is everywhere in the spectrum - it is not a single signal to be thrown away. Its power spectral density (PSD) is flat. So filtering out one image does not reduce the PSD in your passband; it just prevents the other image from contaminating it.

Case 2: Both images pass (folding)

If the IF filter is wide enough that both images fall into the same band, then noise from both sidebands overlaps at the same IF frequencies. The two noise contributions then add:

[ S_{n,\text{out}}(f) = N_0 + N_0 = 2N_0 ]

which is an increase of $+3 dB$ for the noise. The integrated noise in bandwidth $B$ doubles:

[ P_\text{noise,out} = 2 N_0 B ]

The signal, however, still maps only once. This yields a SNR degradation by 3 dB (i.e. a halfing of your signal to noise ratio). This is the classic 3 dB folding penalty seen in zero-IF receivers or in multi-stage down-conversion when both images are allowed.

Phase Noise and LO Noise

On top of thermal noise, mixers also contribute to noise via the imperfections of the LO:

• LO phase noise: sidebands around the LO act like a spreading function. Any strong RF carrier (including blockers) gets smeared into the IF band. At zero-IF, close-in phase noise lands directly into baseband.
• LO amplitude noise: ideally canceled by double-balanced topologies, but leakage can convert it into low-frequency noise and DC offsets.

In sensitive receivers, LO phase noise often dominates the achievable noise floor, even if the thermal SNR is favorable.

Homodyne vs Heterodyne (Recap)

Before looking into the different noise behaviours of homodyne and heterodyne measurement setups for small signals - for example like they are employed in nuclear magnetic resonance (NMR) or electron spin resonance (ESR) experiments let’s take a quick look at the two different configurations:

A Homodyne performs direct conversion. The local oscillator (LO) is operating at the carrier frequency ($f_\text{RF}$). The output is centered at DC. If you use a real mixer, both sidebands fold into baseband which increases the noise by $3 dB$ at this stage. Using I/Q mixers avoids folding.

A Heterodyne operates the local oscillator (LO) at an offset to the carrier frequency $f_\text{RF}$ which yields a non zero intermediate frequency. With a bandpass filter one can keep only one sideband and sees no folding. In case the IF filter passes both images like when performing a second stage that downmixes to DC one experiences folding and thus a $3 dB$ loss of SNR.

We’ve already covered these architectures in detail in another blog post; here we only highlight the noise implications.

Measurements

The following shows an example test setup. It was tested in both configurations with a weak signal:

• Input: 50 MHz at about –110 dBm, pre-amplified by a +48 dB LNA chain. This design choice was taken to match an actual physics experiment.
• Filters: 21.4 MHz bandpass for IF in heterodyne case, 1.2 MHz low-pass on output in both cases on the resulting output.
• Mixers: Mini-Circuits ZLW-2 and ZRDP-1.
• Output frequency: 1 MHz in both setups.

The (near) homodyne configuration

Configuration summary: 50 MHz RF → 49 MHz LO → 1 MHz output → 1.2 MHz LP.

• Input signal is at 50 MHz.
• LO is at 49 MHz.
• The difference product is 1 MHz. This is why I call it near homodyne since we do not directly downmix to DC. This has been done to use an existing measurement setup that was not capable of performing measurements below $9 kHz$.
• The sum product is 99 MHz.
• A 1.2 MHz low-pass is used at the output

The low pass filter removes the 99 MHz sideband completely. Only one image survives. The 1 MHz output signal sees only the noise that was sitting around the 50 MHz RF signal, translated down. No folding happens.

Experimental data

The heterodyne configuration

Configuration summary: 50 MHz RF → 28.6 MHz LO → 21.4 MHz IF (BP filter) → 20.4 MHz LO → 1 MHz output → 1.2 MHz LP.

First stage

• Input signal is at 50 MHz
• First stage LO is set to 28.6 MHz
• The difference produces two sidebands, one at $f_\text{RF}-f_\text{LO} = 21.4 \text{MHz}$, one at $f_\text{RF}+f_\text{LO} = 78.6 \text{MHz}$

One has to consider though that for an input near 50 MHz, there are two possible RF bands that could map to 21.4 MHz IF. Once the desired 50 MHz band but also the image band at 7.2 MHz. In this case one can simply filter the image band to prevent noise contributions.

Second stage

• Input signal is at 21.4 MHz (the intermediate frequency)
• The local Oscillator operates at 20.4 MHz (LO)
• The mixer thus generates:
  • $f_\text{IF} - f_\text{LO} = 1 MHz$
  • $f_\text{IF} + f_\text{LO} = 41.8 MHz$ (that is filtered by a low pass)

One has to consider again that for an desired output of 1 MHz and a LO os 20.4 MHz there are two sidebands that map to the same output frequency of 1 MHz:

• 21.4 MHz via $f_\text{LO} + 1 MHz$
• The image frequency at 19.4 MHz via $f_\text{LO} - 1 MHz$

Depending on the intermediate filter both sidebands fold onto the same 1 MHz sideband. Due to the small difference those are usually not filterable the same way - thus noise doubles (the PSD doubles). The signal on the countrary does not double since only one actual tone maps to the output sideband. This means that SNR drops by 3 dB

Comparison

In the homodyne to 1 MHz case, the low-pass (LP) filtering ensures that only one image survives (i.e. no folding takes place). In the heterodyne first stage, the 21.4 MHz bandpass selects only one image (no folding). But in the heterodyne second stage, the filter is usually wide enough that both sidebands (19.4 MHz and 21.4 MHz) go into the second mixer. At 1 MHz output, they land on top of each other, thus the occuring folding induces a $-3 dB$ drop of SNR.

Possible countermeasures

If one had used an I/Q demodulator or I/Q mixer at the second stage (two mixers with LO phases shifted by 90° against each other, outputs combined as complex baseband), then you’d be able to distinguish +1 MHz from -1 MHz at baseband. Thus folding would not occur, and the $-3 dB$ SNR loss would vanish.

Conclusion

Mixers shift frequencies, but they also influence how noise is mapped:

• Noise PSD is invariant under frequency translation.
• If you reject the image, SNR is preserved (apart from conversion loss).
• If both images overlap (folding), noise doubles because noise is present in the desired and the image band but signals are only present in the desired band; SNR degrades by 3 dB.
• Phase noise of the LO can further raise the noise floor which is often the dominant source of noise.

This article is tagged: Electronics, How stuff works, Basics, Tutorial, DIY