Receiver Noise Figure

 Introduction

The Applet calculates the effective Noise Factor for up to 4 receiver chain components.

Component values may be modified and the display updated using the keyboard 'Enter' Button.

Gains can be negative (attenuator) where the component noise figure is forced numerically equal to the attenuation in dB.

 Background

1) Noise Figure Preamble

The concept of noise figure or noise factor was introduced to describe the excess noise over resistive thermal noise that an active device introduces between its input and output. The definition has various forms but sometimes causes confusion when applied to attenuators or mixers. In this instance it is constructive to be aware of the principle of noise figure measurement instruments. These invariably measure the effective noise figure of a device by injecting a noise source at the input until the noise power measured at the output for example, doubles. The ratio of the injected noise to kTB (Boltzmanns constant x standard temperature-290K x measurement bandwidth) is the indicated noise figure.

For the successful application of the method described in this note it is important to understand the implications of measured and effective noise figure on the noise power available at the output of various components.

2) Signal and Noise Performance of Basic RF Components

2.1) Antenna

The equivalent signal level at the antenna terminal So(f) is then,

So(f) = Si(dBmi) + 10 log {Ga(f)} dBm

where Ga(f) is the antenna gain at frequency f.

Cosmic noise can usually be neglected in the normal range of ESM system sensitivities with the result that the antenna output noise power No(f) is approximately the thermal noise contribution or,

No(f) = kTB

2.2) RF Amplifier

At any frequency 'f' an RF amplifier can be described by it's power gain G(f) and it's noise figure F(f), then the noise power measured at the amplifier output over a bandwidth B centred on f is given by G(f) kTBF(f).

With input signal and noise powers Si(f), Ni(f)kTB at frequency f, where Ni(f) is defined in multiples of kTB, the output signal and noise powers within a bandwidth B centred on f are So(f), No(f) where:-

So(f) = G(f) Si(f)

No(f) = {(Ni(f) - 1) + F(f)}G(f)kTB

The signal equation is obvious, but the noise derivation may require some explanation. Thermal noise generated in the input circuit resistance is included in the noise figure term, F(f) whilst Ni(f)kTB describes the total input noise, so only the excess input noise (Ni(f)-1)kTB is amplified by the gain function G(f).

For example, without any excess input noise the amplifier output noise power is equal to:-

kTBG(f)F(f)

Applying the noise figure measurement principle:-

{(Ni(f) - 1)+F(f)}G(f)kTB = 2kTBG(f)F(f)

then, as defined, the effective noise figure is simply,

Fe(f) = Ni(f) - 1 = F(f)

If the amplifier is noiseless, just kTB extra input noise is required to double the amplifier output noise power, then Ni(f) = 2 and F(f) = 1.

2.3 RF Attenuator/Lossy Filter

A lossy passive component or resistive attenuator having a frequency dependent loss transfer function L(f) produces no noise in excess of thermal at either input or output. When driven with signal and noise powers Si(f), Ni(f)kTB, the outputs So(f),No(f) are

So(f) = L(f) Si(f)

No(f) = (Ni(f) - 1)L(f)kTB + kTB

Showing again that only excess noise at the input is operated on by the transfer function.

The effective noise figure Fe(f) of an attenuator measured by a noise figure instrument is derived from:-

(Ni(f) - 1)L(f)kTB + kTB = 2kTB

or Fe(f) = Ni(f) -1 = 1/L(f)

alternatively, the effective noise figure of an attenuator is numerically equal to the attenuation in dB.

2.4) Power Combiner

Multiport power combiners can be regarded as summing attenuators. If the combining loss from each of 'n' inputs to the output is Ln(f), the output signal and noise components from the 'n' inputs driven by Si,n(f), Ni,n(f)kTB are:-

So(f) = [Sn(Ln(f)Si,n(f))]2

No(f) = Sn(Ln(f)(Ni,n-1)kTB) + kTB

- note that coherent signals (same frequency) sum by voltage and incoherent or noise sources sum by power combination.

2.5) Power Splitter

A multi-way power splitter or divider with loss Ln(f) from the input to the nth output port when driven by signal and noise powers Si(f), Ni(f)kTB produces outputs,

So,n(f) = Ln(f)Si(f)

No,n(f) = Ln(f)(Ni-1)kTB + kTB

2.6) RF Mixer

A diode-based RF mixer generates noise slightly in excess of thermal due to bias currents circulating in the circuit. These currents arise from either intentional d.c. bias to the diodes to reduce mixer local oscillator power requirements or from the local oscillator signal itself. Sometimes termed noise temperature ratio (NTR) with typical values of between 1 to 1.5, it is denoted here by Fd(f). The converted signal at IF is usually significantly smaller than the input RF and this factor is termed the conversion loss Lc(f).

With input signal and noise Si(fs), Ni(f)kTB and a local oscillator frequency fe, the output signal and noise components at IF (one sideband considered - i.e upconversion) are:-

So(fs+fe) = Lc(fs)Si(fs)

No(f+fe) = {(Ni(f) -1)Lc(f)+Fd(f+fe)}kTB

The single sideband noise figure conventionally measured is obtained from

No(f+fe) = 2Fd(f+fe)kTB and,

It is important in the system design to ensure that only front end RF noise from the wanted band is converted to the IF band, either by the use of suitable RF filtering or the use of good performance image rejection mixers. Otherwise, the output signal to noise ratio and effective noise figure will be further degraded.

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