# Determine RF Receiver Specifications

This example shows how to determine the noise figure (NF) and gain (G) of the RF receiver so that you can design a receiver that meets the specifications. This examples uses a 10-bit ADC and target -2.7 dB SNR to achieve the desired BER 1e-4. The example also shows how to compute the chip error rate (ChER) and verify the bit error rate (BER) using the transmitter and receiver you modeled in the Develop ZigBee-Like Baseband Transmitter and Receiver example.

### Determine RF Receiver Noise Figure (NF) and Gain (G)

This section uses traditional heuristic derivations to determine the high-level specifications of the RF receiver based on the given ADC specs.

B = 4 MHz, where B is the simulation bandwidth to use for the signal-to-noise ratio (SNR) calculation.

kT = -174 dBm/Hz, where kT is the thermal noise floor power density at the input port of the receiver.

S = -100 dBm, where S is the sensitivity of the receiver.

SNR = -2.7 dB.

From the receiver SNR, derive the noise power at sensitivity and within the simulation bandwidth: Calculate the noise power at sensitivity (N).

N = S – SNR = – 100 dBm – (– 2.7 dB) = – 97.3 dBm

Calculate the noise power in simulation bandwidth.

N = kT + 10*log10(B) + NF

Solving for the receiver noise figure (NF):

**NF = – 97.3 dBm + 174 dBm/Hz – 10*log10(4e6) = 10.7 dB**

Derive the receiver gain using the ADC specifications.

Nbits = 10, where Nbits is the ADC number of bits

Psat = 0 dBm, where Psat is the ADC saturation power

The simulation bandwidth (B) could be different from the ADC sampling clock frequency (Fadc), for example to model under-sampling ADCs. This example assumes them to be identical.

Fadc, where Fadc is the ADC sampling frequency and which is equal to the simulation bandwidth (Fs)

Nin = N + G + 10log10(Fadc/Fs) = G – 97.3 dBm. This is the receiver noise power in ADC bandwidth

The ADC discretization determines the quantization signal-to-noise ratio (SNRq). From it, derive the quantization noise floor (Nq) that also determines the ADC dynamic range:

SNRq = 6.02 * Nbits + 1.76 dB = 62dB

Nq = Psat - SNRq = 0 dBm - 62 dB = -62 dBm

Assuming an additional 0.1dB contribution to the SNR due to other ADC noise sources, SNRout = SNRin + 0.1dB. The additional noise power due to the ADC can be expressed as:

Nin + Nadc = Nin * 10^(0.1/10) = Nin * 1.0233 (linear scale)

Nadc = Nin – 16dB = G – 97.3dBm – 16dB (in logaritmic scale)

By matching the ADC additional noise and the ADC quantization noise to be at the same level, derive the minimal RF receiver gain (G) that allows meeting the BER specs:

Nadc = Nq = G – 97.3 dBm - 16 dB = – 62 dBm

Solving for the receiver gain (G):

**G = – 62 dBm + 97.3 dBm + 16 dB = 51.5 dB**

### Simulate RF Receiver BER and ChER

Simulate an idealized baseband model of the RF receiver and verify the preliminary RF receiver specifications (NF = 10.7 dB and receiver gain = 51.5 dB). This can be done by collecting 100 errors. The model also computes the corresponding ChER around 7%.

By computing ChER, you can run the subsequent models for less time and still accurately verify the RF receiver specifications. The bottom branch of the model allows you to verify that the receiver SNR matches the original specifications. The ChER difference between top and bottom branch is relatively small, and due to the approximation of the heuristic calculations used to compute the chain gain and NF equivalent to the given SNR.

```
model = 'TopDownRFReceiverDesignIdealBaseband';
open_system(model);
sim(model);
```