How to calculate a mixer noise?

Question

I am struggling to calculate a mixer output noise, but I am stuck in the understanding how to calculate it formally, could someone please help with reasoning or give a reference? I assume that we use the device as up convertor. The goal is to understand the effect of LO and IF to the output, as far as I know that it should be a contribution of a product and LO and IF. Anyway, I use the following approach by assuming the signals (there might be a better approach, I do not know):

$$ S_{IF} = (A_{IF} + A_{IF,N}) cos(\omega_{IF}t +\phi_{IF,N}) + N_{IF} $$ $$ S_{LO} = (A_{LO} + A_{LO,N}) cos(\omega_{LO}t +\phi_{LO,N}) + N_{LO} $$

where $A_{IF}$, $A_{LO}$ are the IF and LO signals, $A_{LO}$, $A_{IF,N}$ are amplitude noises of the IF and LO and $\phi_{IF,N}$, $\phi_{LO,N}$ are phase noises. Finally $N_{IF}$, $N_{LO}$ are some random noises (might be a white noise).

Then if we want to know a output signal it will be: $$ S_{RF,out} = \alpha S_{IF} \cdot S_{LO} $$

Next, I can calculate all the products and so on... However, in order to calculate the signal power (something close to what we measure with a spectrum analyzer: in dBm for signal and dBm/Hz for noise), should I use a spectrum density , I am not sure? Something like average of $ < S_{RF,out}^2 >$, also how can I treat the random variables $A_{IF,N}$, $\phi_{IF,N}$ etc? Should there a correlation of the noise appear somehow?

Answer 1

For a single passive mixer with a real IF and LO inputs, and real RF output, here are the considerations for the noise sources at each input port:

LO input:

In typical application, the LO is driven stronger to forward and reverse bias fast responding (Schottky) diodes, in that they can be approximated as a switch, hard limiting the LO input signal. This has the effect of removing or supressing all AM noise that is present on the LO.

The phase noise as PM noise however passes through and convolves with the signal on the IF input (given the property that multiplication in the time domain which is what a mixer does is convolution in the frequency domain). If the input on the IF port was a single tone with much higher spectral purity (better phase noise), for example, then then output would have the same phase noise in dBc/Hz as the LO input. Phase noise is a spectral density that is non-white so must be considered over specific frequency offsets of concern (application dependent) for its effect on the resulting SNR of the signal. Similarly the phase noise component of the noise floor on the LO port (a white noise floor is half AM and half PM) would translate to the output signal with the same power level relative to the LO signal (dBc): So if for example the broadband LO noise floor degraded such that it was only -50 dBc/Hz, and if the IF noise floor was sufficiently lower to be insignificant), then the PM component of the noise floor on the LO port would be 3 dB lower, and from that we can predict the output noise floor to be -53 dB below the power of the IF signal.

IF Input

There are three primary considerations for noise translation from the IF port to the RF output. One is the conversion loss of the mixer at the LO power operating condition, another is the filtering (if properly provided) of the alias frequency, and the third is intermodulation distortion. The conversion loss of the mixer (typical values are 6-7 dB) attenuates the signal without adding additional noise, but this pushes the signal level closer to the thermal noise floor. This typically wouldn't be a challenge in a transmitter application but a cascaded noise figure computation would use the conversion loss (and assume proper image filtering) to predict the noise at the mixer RF output. (The "Noise Figure" of any device is the difference between the SNR at the device input and the SNR at the device output and takes into consideration the noise relative to the thermal noise floor on each side of the device as well as noise added and signal level lost-- in the case of a passive mixer there is no noise added of any significance but we do have signal lost due to the conversion loss. For more details on cascaded noise figure please see these other posts (in this regard the mixer acts just like a passive attenuator, with frequency translation):

Is noise figure dependent on input noise power?

noise floor of attenuator

Detection Bandwidth for Noise Power Calculation

Anti-Alias filtering is not at all challenging in a transmitter with a high ratio between the RF output frequency and the IF input frequency but must be considered when determining noise. This "filtering" may be partially or completely accomplished via the operational bandwidth of the mixer itself. The RF output of a mixer is the sum and difference of the frequencies at the IF and LO ports, and therefore there are two solutions for frequency bands that can translate from the IF port to the RF port: the primary band of interest and an "alias" band. If the noise floor (as a spectral density) at the IF port was the same in the desired band and alias band, then the noise floor at the RF output of the mixer would be degraded by 3 dB (doubled). It is typically sufficient to ensure that the noise in the alias band is at least 15 to 20 dB below the noise in the primary band of interest.

For example, if we were to frequency translate a 100 MHz IF to a 900 MHz RF using an 800 MHz LO, we would get the desired 900 MHz output as the sum of those two signals, and a 700 MHz output which is an image that also needs to be filtered out after the RF output of the mixer - but that doesn't effect the inband noise we are considering here. In addition, a 1.7 GHz input at the IF port would also translate to a sum frequency of 2.5 GHz and a difference frequency of 900 MHz, our transmitted band of interest. It is likely that the mixer that is used for a 100 MHz IF will have significant loss at 1.7 GHz, but this needs to be confirmed as well as what the noise may be in this band at the mixer IF input.

Additionally non-linearities in the mixer can produce other intermodulation products, for which frequency planning is often done with the help of a mixer spur chart to choose LO and IF frequency band-centers and bandwidths to ensure spurious free operation.