Mixer

Model mixer in RF systems

• Library:

• RF Blockset / Circuit Envelope / Elements

• 

Description

The Mixer block performs signal frequency translation and nonlinear amplification.

For a given RF input signal, V_RF = A_RFcos(ω_RFt) and an LO input signal V_LO = Ac_LOcos(ω_LOt), the mixer multiplies the signals at the input ports:

$$\begin{array}{c}{V}_{\text{in}}{V}_{\text{LO}}=A_{\text{in}}\cos \left({\omega }_{\text{in}}t\right)A_{\text{LO}}\cos \left({\omega }_{\text{LO}}t\right)\\ =\frac{A_{\text{in}}{A}_{\text{LO}}}{2}\cos \left[\left({\omega }_{\text{in}}+{\omega }_{\text{LO}}\right)t\right]+\frac{A_{\text{in}}{A}_{\text{LO}}}{2}\cos \left[\left({\omega }_{\text{in}}-{\omega }_{\text{LO}}\right)t\right]\end{array}$$

This mixing converts the frequency of RF signal to ω_RF + ω_LO and ω_RF – ω_LO. For the mixer to perform this operation correctly, you must include the frequencies ω_RF + ω_LO or ω_RF – ω_LO in the simulation frequencies the Configuration block calculates.

The Power gain specification for this block relates the power of a single-sideband (SSB) to the input.

After mixing the RF and LO signals, the mixer block performs amplification. To model linear amplification, the mixer scales the signals by the coefficient a₁. A Voltage Controlled Voltage Source (VCVS), specified with a polynomial, implements nonlinear amplification. The polynomial includes saturation automatically and produces additional intermodulation frequencies.

Parameters

Main

Source of conversion gain — Source parameter of conversion gain

Available power gain (default) | Open circuit voltage gain | Polynomial coefficients

Source parameter of conversion gain, specified as one of the following:

• Available power gain — The block uses the value of the Available power gain parameter to calculate the linear voltage gain term of the polynomial VCVS, a₁. This calculation assumes a matched load termination for the mixer.

• Open circuit voltage gain — The block uses the value of the Open circuit voltage gain parameter as the linear voltage gain term of the polynomial VCVS, a₁.

• Polynomial coefficients — The block implements a nonlinear voltage gain according to the polynomial you specify. The order of the polynomial must be less than or equal to 9 and the coefficients are ordered in ascending powers. If a vector a has 10 coefficients, [a₀, a₁, a₂, …, a₉], the polynomial it represents is V_out = a₀ + a₁ V_in + a₂ V_in²+ ⋯ + a₉ V_in⁹. In this case, a₁ represents the linear gain term, and the modeling of higher-order terms is done according to [1].

  For example, the vector [a₀, a₁, a₂, a₃] specifies the relation V_out = a₀ + a₁ V_in + a₂ V_in² + ⋯ + a₃ V_in³.

  Trailing zeroes are omitted: if a₃ = 0, [a₀, a₁, a₂] defines the same polynomial as [a₀, a₁, a₂, 0]. The default value of this parameter is [0 1], corresponding to the linear relation V_o = V_i.

Available power gain — Linear gain of mixer

0 dB (default) | scalar

Linear gain of mixer, specified as a scalar in dB. Specify the units from the corresponding drop-down list.

Dependencies

To enable this parameter, select Available power gain in Source of conversion gain tab.

Open circuit voltage gain — Open circuit voltage gain

0 dB (default) | scalar

Open circuit voltage of mixer, specified as a scalar in dB. Specify the units from the corresponding drop-down list.

Dependencies

To enable this parameter, select Open circuit voltage gain in Source of conversion gain tab.

Polynomial coefficients — Order of polynomial

[0 1] (default) | vector

Order of polynomial, specified as a vector.

The order of the polynomial must be less than or equal to 9. The coefficients are ordered in ascending powers. If a vector has 10 coefficients, [a0,a1,a2, ... a9], the polynomial it represents is:

V_out = a₀ + a₁V_in + a₂V_in² + ...  + a₉V_in⁹

where a₁ represents the linear gain term, and higher-order terms are modeled according to [1].

For example, the vector [a0,a1,a2,a32] specifies the relation V_o = a₀ + a₁V₁ + a₂V₁² + a₃V₁³. Trailing zeroes are omitted. If a₃ = 0, then [a0,a1,a2] defines the same polynomial as [a₀,a₁,a₂,0]. The default value of this parameter is [0,1], corresponding to the linear relation V_o = V_i.

Dependencies

To enable this parameter, select Polynomial coefficients in Source of conversion gain tab.

Input impedance (Ohm) — Input impedance of mixer

50 (default) | scalar

Input impedance of mixer, specified as a scalar.

Output impedance (Ohm) — Output impedance of mixer

50 (default) | scalar

Output impedance of mixer, specified as a scalar.

LO impedance (Ohm) — Impedance at LO port of mixer

inf (default) | scalar

Output impedance of mixer, specified as a scalar.

Noise figure (dB) — Single-sideband IEEE noise figure of mixer

0 dB (default) | scalar

Single-sideband noise figure of mixer, specified as a scalar according to the IEEE^® definition.

To model noise in circuit envelope model with a Noise, Amplifier, or Mixer block, you must select the Simulate noise check box in the Configuration block dialog box.

The IEEE SSB definition assumes that the noise in the image bandwidth at the input of the mixer is perfectly rejected, while the mixer internally generates noise in both the image bandwidth and the signal bandwidth. As a result, the noise at the output of the mixer is the sum of two contributions:

$$N_{\text{out}}=N_{\text{in}}{G}_{\text{mix}}+2N_{\text{mixer}}{G}_{\text{mix}},$$

where:

• N_out is the noise at the output of the mixer.

• N_in is the noise at the input of the mixer (assuming that the noise in the image bandwidth is perfectly rejected).

• N_mixer is the noise internally generated by the mixer in both the signal and the image bandwidths.

• G_mix is the mixer gain.

As a result, the noise factor according to the IEEE SSB definition can be expressed as

$$F_{\text{SSB-IEEE}}=1+2N_{\text{mixer}}/N_{\text{in}},$$

which is related to other commonly used definitions through

$$\begin{array}{c}{F}_{\text{SSB}}=2+2N_{\text{mixer}}/N_{\text{in}}=1+F_{\text{SSB-IEEE}},\\ F_{\text{DSB}}=1+N_{\text{mixer}}/N_{\text{in}}=\frac{1}{2}\left(1+F_{\text{SSB-IEEE}}\right).\end{array}$$

You can apply the IEEE SSB definition directly to describe mixer stages when using the Friis formulas for link budget analysis. Using the other definitions requires changing the Friis formulas. Both the Mixer block in RF Blockset™ and the RF Budget Analyzer app in RF Toolbox™ use the IEEE definition.

The analytic relationships between the three definitions allow you to simulate the noise level at the output of the mixer. For example,

$$\begin{array}{c}{F}_{\text{SSB}}=4\text{ dB}\Rightarrow F_{\text{SSB-IEEE}}=10{\log }_{10}({10}^{F_{\text{SSB}}/10}-1)=1.79\text{ dB},\\ F_{\text{DSB}}=4\text{ dB}\Rightarrow F_{\text{SSB-IEEE}}=10{\log }_{10}(2\times {10}^{F_{\text{DSB}}/10}-1)=6.04\text{ dB}\text{.}\end{array}$$

If you simulate an RF Blockset mixer without including an ideal image rejection filter, then the noise at the output of the mixer is larger than that predicted by the noise figure, because the noise in the image bandwidth is effectively folded onto the output signal.

For this reason, when generating models, both the Modulator and Demodulator blocks insert an ideal image-rejection filter automatically. (You can remove the filtering within the block mask.)

The Noise Figure Testbench block measures the SSB noise figure and enables you to verify that the simulated noise figure has the expected value.

• If you add an ideal image rejection filter to your model, then the effective noise figure is consistent with the analytic value.

• If you remove the image-rejection filter, or if you use a filter with partial rejection, then the measured noise figure is larger than the analytic value.

Ground and hide negative terminals — Ground RF circuit terminals

on (default) | off

Select this parameter to internally ground and hide the negative terminals. To expose the negative terminals, clear this parameter. By exposing these terminals, you can connect them to other parts of your model.

By default, this option is selected.

Nonlinearity

Nonlinear polynomial type — Polynomial nonlinearity

Even and odd order (default) | Odd order

Polynomial nonlinearity, specified as one of the following:

• Even and odd order: When you select Even and odd order, the amplifier can produce second- and third-order intermodulation frequencies in addition to a linear term.

• Odd order: When you select Odd order, the amplifier generates only odd order intermodulation frequencies.

  The linear gain determines the linear a₁ term. The block calculates the remaining terms from the specified parameters. These parameters are IP3, 1-dB gain compression power, Output saturation power, and Gain compression at saturation. The number of constraints you specify determines the order of the model. The figure shows the graphical definition of the nonlinear amplifier parameters.

  

Intercept points convention — Intercept points convention

Output (default) | Input

Intercept points convention, specified as Input-referred or Output-referred. Use this specification for the intercept points, 1-dB gain compression power, and saturation power.

IP2 — Second-order intercept point

infdBm (default) | scalar

Second-order intercept point. specified as a scalar. The default value inf dBm corresponds to an unspecified point.

Dependencies

To enable this parameter, select Even and odd order in Nonlinear polynomial type tab.

IP3 — Third-order intercept point

inf dBm (default) | scalar

Third-order intercept point, specified as a scalar. The default value inf dBm corresponds to an unspecified point.

1-dB gain compression power — 1-dB gain compression power

infdBm (default) | scalar

1-dB gain compression power, specified as a scalar. The 1-dB gain compression point must be less than the output saturation power.

Dependencies

To enable this parameter, select Odd order in Nonlinear polynomial type tab.

Output saturation power — Output saturation power

infdBm (default) | scalar

Output saturation power, specified as a scalar. The block uses this value to calculate the voltage saturation point used in the nonlinear model. In this case, the first derivative of the polynomial is zero, and the second derivative is negative.

Dependencies

To enable this parameter, select Odd order in Nonlinear polynomial type tab.

Gain compression at saturation — Gain compression at saturation

infdBm (default) | scalar

Gain compression at saturation, specified as a scalar.

Dependencies

To enable this parameter, select Odd order in Nonlinear polynomial type tab and set Output saturation power .

Model Examples

Validating IP2/IP3 Using Complex Signals

Use the RF Blockset™ Circuit Envelope library to run a two-tone experiment that measures the second- and third-order intercept points of an amplifier. The model computes the intercept points of the amplifier from the modulated signal power measured on each carrier, verifying the behavior of the RF Blockset system. These values are confirmed using the RF Budget Analyzer app and a measurement testbench.

Impact of Thermal Noise on Communication System Performance

Use the RF Blockset Circuit Envelope library to model the thermal noise of a super-heterodyne RF receiver and measure its effects on the bit error rate (BER) of a communications system. The RF receiver model does not include impedance mismatches or nonlinearities. A Communications Toolbox reference model with parameters computed using Friis equations is used to verify the results.

References

[1] Grob, Siegfried, and Jürgen Lindner. “Polynomial Model Derivation of Nonlinear Amplifiers.” Department of Information Technology, University of Ulm, Germany.