## Model System Noise Figure

RF receivers amplify signals and shift them to lower frequencies. The receiver itself introduces noise that degrades the received signal. The signal-to-noise ratio (SNR) at the receiver output ultimately determines the usability of the receiver.

The preceding figure illustrates the effect of the receiver
on the signal. The receiver amplifies a low-power RF signal at the
carrier *f _{RF}* with a high
SNR and downconverts the signal to

*f*. The noise figure (NF) of the system determines the difference between the SNR at the output and the SNR at the input:

_{IF}$$SN{R}_{out}=SN{R}_{in}-N{F}_{sys}$$

where the difference is calculated in decibels. Excessive noise figure in the system causes the noise to overwhelm the signal, making the signal unrecoverable.

### Create a Low-IF Receiver Model

The model `ex_simrf_snr`

simulates a simplified IF receiver architecture. A
Sinusoid block and a Noise block model a two-tone input
centered at *f _{RF}* and low-level thermal
noise. The RF system amplifies the signal and mixes it with the local oscillator

*f*down to an intermediate frequency

_{LO}*f*. A voltage sensor recovers the signal at the IF.

_{IF}To open this model, at MATLAB^{®} command line, enter:

addpath(fullfile(docroot,'toolbox','simrf','examples')) ex_simrf_snr

The amplifier contributes 40 dB of gain and a 15-dB noise figure, and the mixer contributes 0 dB of gain and a 20-dB noise figure, which are values characteristic of a relatively noisy, high-gain receiver. The two-tone input has a specified level of .1 μV. A 1-V level in the local oscillator ensures consistency with the formulation of the conversion gain of the mixer.

To run the model:

Open the model by clicking the link or by typing the model name at the Command Window prompt.

Click

**Run**.

#### Set Up the RF Blockset Environment

To maximize performance, the **Fundamental tones** and **Harmonic
order** parameters specify the simulation frequencies explicitly
in the Configuration block:

*f*, the frequency of the LO in the first mixing stage, equals 1.9999 GHz. and appears in the list of fundamental tones as_{LO}`carriers.LO`

.*f*, the carrier of the desired signal, equals 2 GHz and appears in the list of fundamental tones as_{RF}`carriers.RF`

.*f*, the intermediate frequency, equals_{IF}*f*–_{RF}*f*. The frequency is a linear combination of the first-order (fundamental) harmonics of_{LO}*f*and_{LO}*f*. Setting_{RF}**Harmonic order**to`1`

is sufficient to ensure this frequency appears in the simulation frequencies. This minimal value for the harmonic order ensures a minimum of simulation frequencies.

Solver conditions and noise settings are also specified for the Configuration block:

The

**Solver type**is set to`auto`

. For more information on choosing solvers, see the reference page for the Configuration block or see Choosing Simulink^{®}and Simscape™ Solvers.The

**Sample time**parameter is set to`1/(mod_freq*64)`

. This setting ensures a simulation bandwidth 64 times greater than the envelope signals in the system.The

**Simulate noise**box is checked, so the environment includes noise parameters during simulation.

#### View Simulation Output

The model uses subsystems with a MATLAB Coder™ implementation of a fast Fourier transform (FFT) to generate two plots. The FFT uses 64 bins, so for a sampling frequency of 64 Hz, the bandwidth of each bin is 1 Hz. Subsequently, the power levels shown in the figures also represent the power spectral density (PSD) of the signals in dBm/Hz.

The Input Display plot shows the power spectrum of the signal and noise at the input of the receiver.

The measured power of each tone is consistent with the expected power level of a 0.1-μV two-tone envelope:

$$\begin{array}{c}{P}_{in}=10{\mathrm{log}}_{10}\left(\frac{{V}^{2}}{2R}\right)+30\\ =10{\mathrm{log}}_{10}\left(\frac{{\left(\frac{1}{2}\cdot \frac{{10}^{-7}}{2}\right)}^{2}}{2\cdot 50}\right)+30=-142\text{\hspace{0.17em}}\text{dBm}\end{array}$$

A factor of 1/2 is due to voltage division across source and load resistors, and another factor of 1/2 is due to envelope scaling. See the featured example Two-Tone Envelope Analysis Using Real Signals for more discussion on scaling envelope signals for power calculation.

The measured noise floor at -177 dBm/Hz is reduced by 3 dB from the specified -174 dBm/Hz noise floor. The difference is due to power transfer from the source to the input of the amplifier. The amplifier also models a thermal noise floor, so although this decrease is unrealistic, it does not affect accuracy at the output stage.

The Output Display plot shows the power spectrum of the signal and noise at the output of the receiver.

The measured PSD of -102 dBm/Hz for each tone is consistent with the 40-dB combined gain of the amplifier and mixer. The noise PSD in the figure is shown to be approximately 50 dB higher at the output, due to the gain and noise figure of the system.

If you have DSP System Toolbox™ software installed, you can replace the MATLAB Coder subsystems with a Spectrum Analyzer (DSP System Toolbox) block.

### Simulating Thermal Noise Floor

Thermal noise power can be modeled according to the equation

$${P}_{noise}=4{k}_{B}T{R}_{s}\Delta f$$

where:

*k*is Boltzmann's constant, equal to 1.38065 × 10_{B}^{-23}J/K.*T*is the noise temperature, specified as 293.15 K in this example.*R*is the noise source impedance, specified as 50 Ω in this example to agree with the resistance value of the Resistor block labeled_{s}`R1`

.Δ

*f*is the noise bandwidth.

To model the noise floor on the RF signal at the resistor, the model includes a Noise block:

The

**Noise Power Spectral Density (Watts/Hz)**parameter is calculated as $${P}_{noise}/\Delta f=4{k}_{B}T{R}_{s}$$.The

**Carrier frequencies**parameter, set to`carriers.RF`

, places noise on the RF carrier only.

### Computing System Noise Figure

To model RF noise from component noise figures:

Select

**Simulate noise**in the RF Blockset Parameters block dialog box, if it is not already selected.Specify a value for the

**Noise figure (dB)**parameter of an Amplifier and Mixer blocks.

The noise figures are not strictly additive. The amplifier
contributes more noise to the system than the mixer because it appears
first in the cascade. To calculate the total noise figure of the RF
system with *n* stages, use the Friis equation:

$${F}_{sys}={F}_{1}+\frac{{F}_{2}-1}{{G}_{1}}+\frac{{F}_{3}-1}{{G}_{1}{G}_{2}}+\mathrm{...}+\frac{{F}_{n}-1}{{G}_{1}{G}_{2}\mathrm{...}{G}_{n-1}}$$

where *F _{i}* and

*G*are the noise factor and gain of the

_{i}*i*th stage, and

*NF*= 10log

_{i}_{10}(

*F*).

_{i}In this example, the noise figure of the amplifier is 10 dB, and the noise figure of the mixer is 15 dB, so the noise figure of the system is:

$$10{\mathrm{log}}_{10}\left({10}^{10/10}+\frac{{10}^{15/10}-1}{10000}\right)=10.0\text{\hspace{0.17em}}\text{dB}$$

The Friis equation shows that although the mixer has a higher noise figure, the amplifier contributes more noise to the system.