# Minimum detectable signal

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In general it is clear that for a receiver to see a signal it has to be greater than the noise floor. To actually detect the signal however, it is often required to be at a power level greater than the noise floor by an amount that is dependent on the type of detection used as well as other factors. There are exceptions to this requirement but coverage of these cases is outside the scope of this artical. This required difference in power levels of the signal and the noise floor is known as the signal to noise ratio (SNR). To establish the minmum detectable signal (MDS) of a receiver we require several factors to be known.

Signal to Noise Ratio (SNR) in dB.

Detection Bandwidth (BW) in Hz.

Temperature of the receiver system To in Kelvin

Boltzmann's constant 1.38e-23 Joules per Kelvin

Receiver Noise Figure (NF) dB

To calculate the minimum detectable signal we first need to establish the noise floor in the receiver by the following equation:

Noise Floor (dBm)=10Log(K.T0/1E-3) + NF + 10Log(BW)

As a numerical example:

A receiver has a bandwidth of 100 MHz and noise figure of 1.5 dB and the physical temperature of the system is 290 Kelvin.

```                  Noise Floor (dBm) = 10Log(1.38e-23 x 290 /1E-3) + 1.5 + 10Log(100e6)
=-174 + 1.5 + 80 (dBm)
=-92.5 (dBm)
```

So for this receiver to even begin to see a signal it would need to be greater than -92.5 dBm. Some confussion often arises because the level calculated above is also sometimes called the Minimum Desernable Signal (MDS). For the sake of clarity we will refer to this as the noise floor of the receiver. The next step is to take into account the SNR required for the type of detection we are using. If we need the signal to be 10 times more powerful than the noise floor the required SNR would be 10 dB. To calculate the actual minimum detectable signal is simply a case of adding the required SNR in dB to the noise floor. So for the example above this would mean that the minmum detectable signal is:

MDS (dBm) = -92.5 + 10 = -82.5 (dBm)

MDS (dBm) = 10Log(kTo/1e-3) + NF + 10Log(BW) + SNR (dBm)

In this equation:

kTo is the available noise power in a bandwidth BW = 1Hz at To, expressed in dBm. To is the system temperature in kelvins and k is Boltzmann's constant (1.38×10−23 joules per kelvin = −228 dBW/(K·Hz)). If the system temperature and bandwidth is 290 K and 1 Hz, then the effective noise power available in 1 Hz bandwidth from a source is −174 dBm (174 dB below the one milliwatt level taken as a reference).

1 Hz noise floor: calculating the noise power available in a one hertz bandwidth at a temperature of T = 290 K defines a figure from which all other values can be obtained (different bandwidths, temperatures). 1 Hz noise floor equates to a noise power of −174 dBm so a 1 kHz bandwidth would generate −174 + 10 log10(1 kHz) = −144dBm of noise power (the noise is thermal noise, Johnson noise).

MDS (dBm) = 10Log(kTo/1e-3) + NF + 10Log(BW) + SNR (dBm)

The equation above indicates several ways in which the minimum detectable signal of a receiver can be improved. If one assumes that the bandwidth and SNR are fixed however by the application, then one way of improving MDS is by lowering the receivers physical temperature. This lowers the NF of the receiver by reducing the internal thermally produced noise. These type of receivers are referred to as Cryogenic Receivers.

## Definitions

### Noise figure and noise factor

Noise figure (NF) is noise factor (F) expressed in decibels. F is the ratio of the input signal-to-noise-ratio (SNRi) to the output signal-to-noise-ratio (SNRo). F quantifies how much the signal degrades with respect to the noise because of the presence of a noisy network. A noiseless amplifier has a noise factor F=1, so the noise figure for that amplifier is NF=0 dB: a noiseless amplifier does not degrade the signal to noise ratio as both signal and noise propagate through the network.

If the bandwidth in which the information signal is measured turns out not to be 1 Hz wide, then the term 10 log10(bandwidth) allows for the additional noise power present in the wider detection bandwidth.

### Signal-to-noise-ratio

Signal-to-noise-ratio (SNR) is the degree to which the input signal power is greater than the noise power within the bandwidth B of interest. In the case of some digital systems a 10 dB difference between the noise floor and the signal level might be necessary; this 10dB SNR allows a bit error rate (BER) to be better than some specified figure (e.g. 10−5 for some QPSK schemes). For voice signals the required SNR might be as low as 6 dB and for CW (Morse) it might extend, with a trained listener, down to 1 dB difference (tangential sensitivity). Usable in this context then means it conveys adequate information for decoding by a person or a machine with acceptable and defined levels of error.

Above definition is wrong. Signal-to-noise-ratio is ratio not between POWER of signal and POWER of noise, but it is ratio between ENERGY of signal to the ENERGY of noise in the frequency band of signal. This value can differ from the ratio of powers on the orders of magnitude