Random Signals and Noise

1. Random Signals Deterministic Systems and Signals can be described by the Frequency Response or Frequency Content respectively. Thus, for example, a 3rd-order lowpass filter: Amplitude response in Nepers Phase response in degrees Furthermore:

Power Spectral Density of x(t ): Random, wide-sense stationary* (wss) signals are best described in the frequency domain by the Power Spectral Density (PSD). According to the Wiener-Khintchine Theorem the PSD of a function is the Fourier Transform of its Autocorrelation function Rxx(t ) of a function x(t ): Power Spectral Density of x(t ): Fourier Transform of Rxx(t ) Total power in signal x(t ) Note: * We assume that the functions are wide-sense stationary (wss) in which case the mean and Rxx(t ) are time independent.

For a wss random signal we have: Thus: Expected Value “mean” Second moment of x. For a wss random signal we have: is the Fourier Transform of and is the power spectral density of . is the expected value or second moment of .

The mean or average value of is the expected value of . The variance of is given by For a zero-mean signal standard deviation of or rms value of The standard deviation provides a measure of how the “mass” of a density function is dispersed about the mean.

2. Random Signals as Input to a Linear Time-Invariant System Impulse response Transfer function Output: Convolution Multiplication Four. Trans: Since is a wss random signal it is more useful to consider the autocorrelation functions and and their Fourier Transforms and , respectively.

We know that and are the power spectral density (PSD) of , , respectively. It can be shown that: or: Taking the FT of we obtain the PSD of the output signal: Taking the FT of we obtain:

The PSD of the output signal is given by the product of the PSD of the input and the squared magnitude of the transfer function of the system. The power transfer function of a system with impulse response follows as: Output Power :

With one-sided PSD: and: For essentially constant input spectral density over frequency band B=f2-f1: where:

Power Gain PG with input PSD which is constant over frequency band B: where: Note: represents frequency response e.g. of a filter; since only occurs, phase characteristics play no part.

Ex: Power Gain/Loss and Component Tolerances For exact component values: With non-exact component values: Relative Change in power gain: RG in dB:

3. Noise Unwanted disturbance superposed upon its information content is summarily designated as noise. One distinguishes between extrinsic noise, i.e. noise originating from sources outside a circuit (e.g. electromagnetic interference or EMS) and intrinsic noise, i.e. noise generated by the circuit itself. In general, noise is considered to be a random signal. (Deterministic disturbances are given other names, e.g. hum). Here we consider only intrinsic noise, or simply noise.

White noise has zero mean, i. e White noise has zero mean, i.e. , and a flat or constant spectral density, i.e. . Thus , where is an impulse or delta function centred at the origin. [Although communication channels are limited mainly by intersymbol interference and impulsive noise, they are generally modelled as channels with additive white Gaussian noise (AWGN), i.e. noise with a Gaussian probability density function and zero mean.]

4. Noise Voltage and Current In most solid-state devices and electrical components a noise voltage or current can be registered, even in the absence of an external source of power. Instead of power spectral density we then speak of voltage or current spectral density or, simply, of noise spectral density. The rms noise voltage or current is related to its noise spectral density by: where the expected value of (or the average value ) is zero. Note that designates a voltage or current divided by . For a limited frequency band of , the limits of integration are the band limits.

5. The Noise in MOSFETs There are three distinct sources of noise in solid state devices: thermal noise, shot noise, and flicker noise. a. Thermal Noise Caused by random thermal motion of the electrons in a resistor R. As a result, a fluctuating voltage vnT appears across the resistor, even in the absence of a current from an external circuit. Rid: “ideal”, noiseless resistor

mean square of VnT: mean square of inT: K: Boltzmann’s constant: 1.38×10-23 J/K T: absolute temp. in degrees Kelvin At T=300°K (27°C): [Ex: ] Df: bandwidth in which noise is measured. Note: (i) average value (dc component) of thermal noise is zero (ii) spectral density is independent of frequency: thermal noise is “white”.

The gate referred noise voltage source is given by: MOSFET in Conduction contains free carriers, therefore is subject to thermal noise. In this case R given by incremental channel resistance If device is in saturation, its channel tapers off, and [Note: we “pretend” that inT is caused by a voltage source vnT connected to the gate of a noiseless MOSFET!] The gate referred noise voltage source is given by:

inT and vnT depend on the dimensions, bias conditions, and temperature of the device. Example: Transistor with W=200mm, L=10mm, Cox=4.34×10-8F/cm2 (corresponds to 800 oxide thickness), drain current in saturation iD=200mA. Gate-referred noise voltage at room temperature: . If device is switched off: R very large, G very small: very small! Thus, for usual (low or moderate) external impedance levels, the MOSFET can be regarded as a noiseless open circuit if it is turned off.

I: nominal (average) current b. Shot Noise Caused by randomly propagating charge carriers (electrons or holes) superimposed on the nominal (average) current I: fluctuation in the number of carriers crossing a given surface in the conductor in any time intervals: ins. I: nominal (average) current q: magnitude of electron charge 1.6×10-19 Coulomb Df: bandwidth. This formula holds only for very low carrier density and high external electric field: interaction between carriers negligible. Otherwise: randomness of density and velocity reduced due to correlation introduced by repulsion of the charges. ins then much smaller than predicted by formula.

In a conducting MOSFET channel , charge density is high, electric field is low. Above formula does not hold. Noise current due to random carrier motions then better described as thermal noise (see above). Shot noise has a flat spectral density up to optical frequencies. c. Flicker (1/f) Noise Extra electron energy states existing at boundary between Si and SiO2 cause trapping and releasing of electrons from channel: relatively slow, low-frequency process, proportional to 1/f, hence 1/f noise. DC value of noise current inf is zero.

d. Channel Noise in MOSFET Possible model of flicker noise is current source in parallel with channel resistance (as above), or gate-referred noise voltage vng. For “clean” process we have: K depends on temp. and fabrication process: typically »3×10-19 V2×F. channel current noise » independent of bias conditions d. Channel Noise in MOSFET Equivalent noise current generator in: this is root mean square noise current resulting e.g. from thermal noise current inT and flicker noise current inf, thus:.

This assumes that the individual noise sources are statistically independent (or, more generally, uncorrelated). In general, total noise xrms due to n independent (or uncorrelated) noise sources xrms(i) is given by: rms summation formula.

Gate-referred voltage source: Summary of Noise Models:

6. Noise Analysis Tools Purpose of Noise Analysis: to determine the equivalent input/output Signal-to-Noise Ratio (SNR) of a circuit. To this end, all noise produced by the various circuit components is thought to be concentrated into a single noise source, the socalled ‘equivalent’ noise source. The power contents of this source equals the noise power produced by the whole circuit. a. Equivalent Noise Source replaces all other noise sources in circuit. Once this source is obtained, the remainder of the circuit can be considered ‘noise free’. It is obtained by transforming all circuit noise sources to a single position.

Example: Note: In general, circuit behaves linearly w.r.t. (small signal) noise; thus time-domain transfer from vn,2i and in,2i+1 to en,eq can be represented by an impulse response hi(t). Furthermore * denotes convolution.

b. Useful Network Transforms for Noise Analysis 1. Voltage-Source Shift: V-shift Transform. SFG 2. Current-Source Shift: I-shift Transform.

3. Norton-Thevenin Transform: I to V Source Transform. 4. Shift through Twoports: Transform. of Noise Sources thru’ a twoport. ; ;

Output noise tends to zero as A=B=C=D® 0, i.e. This is the [A] matrix of a NULLOR. Thus the ideal gain element is a Nullor (inf. gain, etc.) Thus: The output noise voltage source vn is transformed into an input noise voltage source Avn and a noise current source Cvn that are fully correlated. Likewise, the output noise current source in is transformed into an input noise voltage source Bin and an input noise current source Din.

c. Impedance Feedback Amplifiers Example: Voltage Amplifier:

7. Noise and Dynamic Range of Active-RC 2nd-Order Biquad Filter Given I-2-DF bandpass filter: Transfer functions from noise source Nx of component x to output Vout:

[Nx is either voltage or current noise source of x!] 1. Find every transfer function from each Nx-to-Vout: iR11, iR12 , iR4 , iR5 , iR6 , en , inp , inn. [Nx is either voltage or current noise source of x!] Dynamic Range Measure DR: (Vout)eff: maximum undistorted RMS output voltage. (En)eff: RMS noise voltage within a specified frequency range. Thus, means-square value is given by:

v2n(w): Square of noise spectral density, derived from all noise sources and corresponding transfer functions. Ti,k: transfer impedance from kth current noise source (in)k to output voltage. Tv,l: voltage transfer function from lth voltage noise source (vn)l to output voltage.

8. Noise Factor / Noise Figure Noise Figure is the Noise Factor converted to dB. These parameters specify quantitatively how noisy a device or amplifier is. The noise figure (NF) is the ratio of the noise power output of the circuit under consideration to the noise power output which would be obtained in the same bandwidth if the only source of noise were the thermal noise of the source resistance Rs, e.g. the NF compares the noise in an actual amplifier with that in an ideal (noiseless) amplifier. It is customary to convert the output power to an equivalent input power, thus noise Factor F:

Intrinsic Noise of device (e.g. amplifier) with power gain PG: Equivalent input noise sources v2n, i2n. Source resistance Rs; thermal noise contributed by Rs: vnT Total Equivalent input noise: v2eq: Total Equivalent output noise: v2no:

Example: RF circuits assume Rs=50W. (Noiseless Device: F=1) Noise Factor: Example: RF circuits assume Rs=50W. Noise Factor F is a power ratio; converted to dB it becomes Noise Figure NF: Note: F=1: amplifying device adds no additional noise to input signal.

Signal to Noise Ratio (SNR): SNR decreases with amplification.

Example: Opamp Input Noise: 1. Noninverting Opamp: No source resistor necessary, therefore Rs=0, vnT=0. =0 =0 Equivalent input noise voltage:

2. Inverting Opamp: Equivalent input noise voltage:

3. Shot Noise: (charge of an electron) : negligible compared with thermal noise!

9. Noise Figure and SNR vnT: Thermal noise associated with internal resistance R(f) of source. Output noise of device is made up (i) contribution due to source and (ii) contribution due to device. Available output noise power in frequency band Df (centered at frequency f): maximum average noise power in Df, obtainable at output of device. It is obtained when: load impedance is complex conjugate of device output impedance:

With: Output Noise Figure* (per unit bandwidth) SNO(f): Spectral density of total noise power at device output. SNS(f): Spectral density of noise power due to source at device input. PG(f): Available power gain: In physical device: F>1 (or, more accurately: NF >1) * we use F instead of NF for the noise figure in [dB] because this is the common designation.

Alternative Expression for F (or NF, respectively) Consider: Let: PS(f): max. average signal power available from source. With: ZL(f)=Z*S(f)=R(f)-jX(f) max available power from source

Thus from above: PO(f): Available signal power at device output. Available SNR of source Available SNR at device output. Both measured in a narrow band Df centered at f.

Since F0 is always greater than unity, it follows that rs>r0, i.e. SNR ALWAYS DECREASES WITH AMPLIFICATION! F0 as defined above is function of operating frequency f and is referred to as SPOT NOISE FIGURE. In contrast: AVERAGE NOISE FIGURE F0: For constant (white) noise (as for thermal noise) at device input, over Df: F0 º F0

10. Equivalent Noise Temperature For low-noise devices, F is close to unity! In such cases equivalent noise temperature is easier to use. Available noise power into device: N1 [ For sinusoidal source, we saw earlier: ] Available noise power: N1 Available noise power: N2 Here:

Total output Noise Power: Nd: Noise power contributed by 2-port device to total available output noise power N2. We define: where PG: available device power gain; Te: equivalent noise temperature. Total output Noise Power: Thus: Thus: Equivalent Noise Temperature. where F0 is the noise figure of the device measured under matched input conditions, and with the noise source at temp T.

11. Noise Figure of Composite 2-port Networks Consider cascade of two noisy, impedance- matched, two-port networks: With input impedance matching we found From above:

Due to impedance matching Rin2=Rout1 we have: From above: Due to impedance matching Rin2=Rout1 we have: Thus: output noise power Noise Figure: output noise power for noiseless devices

For the matched cascade of any number of two-port networks: If the first stage of the cascade has a high gain, the overall noise figure F0 is practically the same as that of the first stage!