Frequency mixers make most of the analog signal processing in RF/microwave receivers possible. They allow translation of high-frequency signals to a lower intermediate frequency (IF) where digital-processing techniques can be applied. Mixers are three-port devices, featuring an RF, local oscillator (LO), and IF port. For mathematical modeling purposes, however, it may help to consider a fourth port.

Modeling mixer noise behavior requires a precise knowledge of the effective input noise temperatures derived from measurement, or determined from manufacturer data. Figure 1 shows the model for a generic mixer.^{1} in addition to the usual three ports (RF, IF, and LO), it includes a fourth port to account for the conversion of the image band.

In the model of Fig. 1, the mixer is represented as a noise-free device with effective input noise temperatures to account for the lossy nature of the conversion process and for any additive noise contributed by the mixing devices. These noise sources are typically lumped into parameter T_{e}, a fictitious resistor with a physical body temperature required to produce the required noise power. Parameter T_{t} is the input noise termination for the mixer.

In this proposed model, term T_{e} can be broken into two parts as shown in **Fig. 2**: T_{eC} and T_{eN}. Term T_{eC} represents the effective input noise temperature corresponding to the conversion loss of the mixer, and is similar in nature to the effective input noise temperature for an attenuator.^{2,3} Parameter T_{eN} represents the effective input noise due to all other internal noise sources.

To model the noise behavior of an attenuator, one adds an effective input noise temperature to a noiseless attenuator, as shown in **Fig. 3**. For the case of a frequency mixer, its conversion loss gives rise to a signal-to-noise degradation that is similar to the behavior of an attenuator, as shown in Eq. 1 for the effective input noise temperature:

T_{e} = 290(loss factor 1) (1)

where the loss factor can be found by Eq. 2:

Loss factor = 10^{(Attn/10)} (2)

In the case of a frequency mixer, the value for T_{eC} is modified slightly due to the contribution of the image channel. For a mixer, term T_{eC} is calculated by means of Eq. 3:

T_{eC} = 145(conversion loss -1) (3)

where conversion loss refers to a numeric quantity and term T_{eC} is the effective input noise temperature accounting for the mixer's conversion loss.

This term accounts for the signal-to-noise degradation at the output from the mixer due to the conversion loss. in the limit, as the conversion loss of the mixer goes to infinity, the output noise should approach KTB or a temperature, T, of 290 K, where K is Boltmann's constant (1.3806488 x 10^{-23} J/K) and B is bandwidth (in Hz).

In determining the noise figure for a mixer, the noise factor is defined by Eq. 4:

Noise factor = (s_{i}/N_{i})/(S_{o}/N_{o}) = (s_{i}/N_{i})(S_{o}/N_{o}) (4)

For a noiseless mixer^{1}:

N_{o} = KBt + T_{eC})G_{cS} + (T_{t} + T_{eC})G_{cI}> (5)

where G_{cS} and G_{cI} are the conversion-gain values for the signal and image channels. The noise factor for a noiseless mixer can be represented by Eq. 6:

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Noise factor (noiseless) = (S_{i}/N_{i}){t + T_{eC}) (G_{cS} + G_{cI})>/(S_{i}G_{cS})} (6)

For the case where G_{cS} = G_{cI}, we have Eq. 7:

Noise factor = 2(T_{t} + T_{eC})/T_{t} (7)

So far, the effective input noise temperature for the conversion-loss term given in Eq. 3 has been determined. The next step is to determine TeN, the effective input noise temperature which accounts for the added noise at the output of the mixer not related to the converted noise in the signal and image channels.

For the model of **Fig. 2**, the output noise is equal to the modification of Eq. 5 shown by Eq. 8:

N_{o} = KBt + T_{eC} + T_{eN})G_{cS} + (T_{t} + T_{eC} + T_{eN})G_{cI}> (8)

Given a measured single-sideband (SSB) noise factor of NFac, an expression for NFac can be written in terms of TeN, and then the expression (Eq. 9) solved for T_{eN}:

NFac = (S_{i}/T_{t})t + T_{eC} + T_{eN})(G_{cS} + G_{cI})>/S_{i}G_{cS} (9)

This yields Eq. 10:

T_{eN} = {cS(NFac)T_{t}>/(G_{cS} + G_{cI})} - T_{t} - T_{eC} (10)

If G_{cS} = G_{cI} , then T_{eN} can be solved by means of Eq. 10:

T_{eN} = {t >/2} - T_{t} - T_{eC} (11)

To validate this model, the Virtual System Simulator (VSS) software from AWR was put to use. The circuit shown in **Fig. 4** was used for the evaluation, which is essentially a noiseless amplifier followed by a mixer. The amplifier's gain was used to vary the level of noise into the mixer. The evaluation involved examining cases from a mixer with conversion loss of 5 dB (**Fig. 5**) and SSB noise figures from 7 to 11 dB, to a mixer with conversion loss of 9 dB (**Fig. 6**) and SSB noise figures from 10 to 14 dB. In all cases, the values predicted by the proposed mixer behavioral model were compared with the results predicted by VSS.

Finally, the noise power spectral density (PSD) of the output noise from the mixer, for the case of a mixer with 6-dB conversion loss and 8-dB noise figure, was used as the basis for comparisons of results from the proposed mixer behavioral model and the VSS predictions (**Fig. 7**). In all cases, the results computed by the proposed model were in good agreement with the values predicted by the VSS simulation software.

One of the unexpected consequences of using a mixer that has not had the image band eliminated is a 3-dB increase in the system noise figure. This result arises when an amplifier precedes the mixer, and care has not been taken to remove the image response by either an image-reject filter or through the use of an image-reject mixer. When designing an LNA-mixer cascade for a receiver system, the first expectation is that the amplifier will set the noise figure of the system according to the familiar noise cascade equation. Unfortunately, the cascade equation does not account for the added noise folded over from the image band into the IF band.

The amplifier preceding the mixer will generate a broadband noise spectrum. When processed by the mixer, the folded-over noise at the IF will double the IF output noise spectrum and cut in half the signal-to-noise ratio (SNR), effectively increasing the system noise figure by 3 dB. The actual amount of SNR degradation will depend on a number of factors, mostly controlled by the gain of the LNA. For small LNA gains, the system noise figure will improve. In the limit, as the gain of the LNA goes to infinity, the noise figure of the system will be equal to the noise figure of the LNA plus 3 dB (**Fig. 8**).

REFERENCES

1. William E. Pastori, "Image and Second Stage Corrections Resolve Noise Figure Measurement Confusion," Microwave System News, May 1983.

2. Jensen and Warnick, "ECEn 464: Wireless Communications Circuits," 3.5.3 Lossy Components, November 18, 2009.

3. Stephen D. Gedney, "Noise, SNR, MDS and Noise Phasors", EE521, pg 17, Set 19, Dept. of Electrical Engineering, University of Kentucky, Lexington, KY.