Fig. 1: One realization of a noise signal.
One way to classify signals is according to their energy and power. Let x(t) denote a signal. Signals with finite energy satisfy
As time approaches positive or negative infinity such signals must approach zero. Periodic signals do not have finite energy. Instead they often satisfy a less strict requirement, i.e. they have finite power.
where T denotes the period of the signal. Noise signals are generated by random processes. We are going to focus on noise signals with finite power. First of all, noise signals are random signals. The same noise source can generate infinitely many realizations of the same noise signal.
Fig. 1: One realization of a noise signal.
Observing the dependence of the signal on time does not deliver much useful information. Let x(t) denote a realization of a noise signal (Fig. 1). All realizations have some common properties. Mathematically these properties can be formulated via the correlation function. The correlation function of two signals x(t) and y(t) is defined as
where E[.] denotes expectation, i.e. the mean value computed across all possible realizations of the two signals. If x(t) and y(t) are generated by two stationary random processes the correlation function depends only on τ. If the two processes are also ergodic (i.e. their statistical properties can be obtained by observing a single realization for a sufficient amount of time) then averaging over all realizations can be replaced with averaging over time.
Fig. 2: Illustration of stationarity and ergodicity. For a stationary random process all sample averages are the same regardles of time at which they are computed. For an ergodic process all sample averages are equal to the time average of every realization of the signal.
Noise signals we meet in practice are ergodic. Therefore
If we choose y(t)=x(t) the correlation function is also referred to as the autocorrelation function.
The Fourier transform of the autocorrelation function is also referred to as the power spectral density.
where f denotes the frequency. The integral of the power spectral density is equal to the signal's power (Parseval's theorem).
It can be shown that the power spectral density is a nonnegative function of frequency. The autocorrelation function is symmetric (cxx(-τ)=cxx(τ)). Therefore the power spectral density is an even function (i.e. Sxx(-f)=Sxx(f)) so it is sufficient to know its values for nonnegative frequencies. With this in mind we introduce one-sided power spectral density.
Parseval's theorem for one-sided power spectral density can be written as
We introduced one-sided power spectral density because it is used for noise characterization in circuit simulators. It also has a physical meaning. Suppose we pass a noise signal x(t) through an ideal bandpass filter with pass-band between f1 and f2, and measure the RMS value (i.e. the root mean square) of the output signal y(t). The filter eliminates all frequencies outside the pass band. The following relation connects the measured RMS value with the power spectral density
This type of noise arises due to the chaotic movement of electrons in the conductor. Every resistor generates thermal noise. It can be modelled with a current source i(t) in parallel with the resistor (Fig. 3, right). Note that the polarity of the source is not important because the power spectral density does not change if we reverse the current.
Fig. 3: Ideal resistor (left) and a resistor that generates thermal noise (right).
The current source produces thermal noise with the following one-sided power spectral density
where h is the Planck constant, k is the Boltzmann constant, T is the absolute temperature, and R is the resistance of the resistor, Because at room temperature the exponent in the exponential term is small for frequencies below 6THz we can simplify the formula to
We can see the power spectral density does not depend on the frequency across a very wide range of frequencies. Therefore we refer to this noise as white noise.
This type of noise occurs because the electric current consists of a flow of discrete charges (electrons). Its power spectral density does not depend on temperature or frequency. In a diode this type of noise is modelled by a current source in parallel with the p-n junction (Fig. 4, right, represented by current source is(t)).
Fig. 4: Ideal diode (left) and a diode with current sources representing flicker noise if and shot noise is (right).
The power spectral density of of the current source representing shot noise is
where q is the electron charge and I is the current flowing across the p-n junction. Shot noise is also "white".
The power spectral density of flicker noise is inversely proportional to the frequency. This type of noise is also referred to as 1/f noise. In a semiconductor diode flicker noise is modelled by a current source in parallel with the p-n junction (Fig. 4, right, represented by current source if(t)). The power spectral density of the current source is
where Kf and Af are two constants that characterize flicker noise, and I is the current flowing through the p-n juncion. Noise signals with power spectral densities proportional to 1/f are also deemed pink noise.
Fig. 5: A current source for modelling channel thermal noise and flicker noise in a MOS transistor.
We can model noise by introducing noise sources in arbitrary semiconductor devices. Take, for instance, an MOS transistor (Fig. 5). The channel thermal noise and flicker noise can be modelled with a single current source connected between the drain and the source terminal. We are not going to introduce the expression for the power spectral density of this source because it exceeds the scope of this lecture. Let us only note that the power spectral density depends on the currents and voltages at the terminals of a MOS transistor, as well as the absolute temperature.
All resistors and semiconductor devices are sources of noise. The noise generated by circuit elements is modelled by current sources. For shot and flicker noise the power spectral density of the noise current source depends on the current flowing through the device. One semiconductor element contributes several such noise sources to the circuit. A diode, for instance, contributes three: shot noise and flicker noise generated by the current flowing across the p-n junction, and thermal noise of the contact resistance. A bipolar transitor has 5 noise sources: shot noise and flicker noise due to the device current, and thermal noise originating from the contact resistances of the emitter, base, and the collector terminals.
Before we proceed to computing the output noise of a circuit we must introduce the notion of a small-signal transfer function. Suppose there is an independent source in the circuit that generates a small-signal sinusoidal excitation characterized by complexor X. Let us assume this complexor does not depend on the frequency. The excitation results in small sinusoidal responses observed all over the circuit. Let Y denote the complexor representing one such response observed at a selected point in the circuit. The small-signal transfer function from the independent source to the selected point where the response is observed is defined as
Note that the transfer function depends on the frequency. It can be computed from the results of the small-signal frequency-domain analysis (i.e. AC analysis in SPICE) by simply dividing the observed response with the complexor representing the excitation. The unit of the transfer function depends on the type of the observed response (voltage or current) and the type of the excitation (independent voltage or current source). If the circuit is excited by a voltage source and the observed response is a current then the units of the transfer function are A/V. If, on the other hand, the response is also a voltage, the transfer function is a dimensionless quantity (since it is a ratio of two voltages).
Nonlinear circuits behave as linear if we consider only the small signal sinusoidal excitations and the corresponding response. The steady-state response of a linear system excited by an independent sinusoidal source is sinusoidal and can be represented by a complexor (Y). This complexor can be computed from the complexor (X) representing the excitation by multiplying it with the corresponding small-signal transfer function.
where f denotes the frequency of the excitation. If the input signal is a small noise signal with power spectral density Sxx+(f) then the output of the linear system is also a noise signal with power spectral density Syy+(f). The output power spectral density can be computed as
Now suppose a linear system is excited by two independent sources X1 and X2. If we observe the response of the system to X1 while X2 is turned off (i.e. set to zero) the response can be expressed as
where H1 is the transfer function from the first independent source to the output of the system. Similarly, if X1 is turned off and the system is excited only by X2 the response is
where H2 is the transfer function from the second independent source to the output of the system. Now suppose the system is excited simultaneously by both independent sources. The response of the system can then be expressed as
This property of linear systems is also referred to as superposition. With this tool in our hands we could compute the time-domain responses of a circuit excited by individual noise sources and then obtain the response of the circuit excited by all noise sources simultaneously by adding up these partial responses. If, however, we are interested in the power spectral density of the response we need one more piece of the puzzle.
So how do we treat noise signals that are obtained by summing two noise signals x1(t) and x2(t)? Things are quite simple if the two signals are uncorrelated, i.e. the correlation function satisfies
In real-world circuits the noise sources representing various types of noise generated by a circuit element are uncorrelated. Similarly, noise sources modelling the noise generated by two distinct circuit elements are also uncorrelated. Let S11+(f) and S22+(f) denote the power spectral densities of x1(t) and x2(t), respectively. Then the power spectral density of y(t)=x1(t)+x2(t) is
Uncorrelated noise signals remain uncorrelated even after they are transformed by a linear system. If we put together all we have learned so far: for a linear system excited by two uncorrelated noise sources with power spectral densities S11+(f) and S22+(f) the power spectral density of the output noise can be expressed as
where H1 and H2 are transfer functions from the two noise sources to the circuit's output. This formula can be generalized to an arbitrary number of noise sources and is the basis for small-signal noise analysis in circuit simulators.
We illustrate the small-signal noise analysis with an example. Take, for instance, the nonlinear circuit in Fig. 4. The circuit includes the noise sources modelling the noise generated by the two resistors and the MOS transistor.
Fig. 6: A nonlinear circuit with noise sources (in1, in2, and in3).
The first step of small-signal noise analysis is the computation of the circuit's operating point. In this computation all noise sources are disabled (set to 0). The obtained operating point is used for computing the power spectral density of the noise generated by the MOS transistor represented by in3. The power spectral densities of the noise sources in1 and in2 are independent of the operating point (thermal noise). The operating point is also used for computing the linearized models of nonlinear elements (i.e. MOS transistor). Let Ugg and Udd denote the complexors respresenting the AC part of the excitation generated by the two independent voltage sources in the circuit. After the linearization is complete the frequency-domain system of equations for the linearized circuit is assembled (see previous lecture).
The procedure we are going to describe computes the power spectral density of the output noise at a single frequency. Let us assume the output signal is the nodal voltage v2. The output noise comprises contributions from all three noise sources. To compute these contributions we first need to compute the transfer functions from every noise source to the output. For that purpose the small-signal analysis described in the previous chapter is used.
To compute the transfer function from in1 to v2 we must construct the corresponding system of equations for a circuit where the only sinusoidal excitation is comming from in1. The coefficient matrix of this system does not depend on the noise source because independent sources contribute only to the right-hand side of the system. The only thing we need to construct is the right-hand side, which is easy, as we already know the matrix footprint of an independent current source. If we set the complexor representing the AC current generated by in1 to 1 while all other independent sources are set to 0 the complexor of the response observed at v2 will be identical to the transfer function we want to compute.
By solving for V2ac we obtain the desired transfer function denoted by H1. Similarly for the transfer function from in2 to v2 the system of equations is
By solving for V2ac we obtain the transfer function denoted by H2. Finally, to obtain the transfer function from in3 to v2 we must solve
By solving for V2ac we obtain the transfer function denoted by H3. Because the matrix of coefficients depends on the frequency (ω) we can compute the values of all required transfer functions at a single frequency with only one LU decomposition which is common to all transfer functions at given ω. Computing the value of a transfer function for one ω requires only one forward and one backward substitution.
Now suppose S11+, S22+, and S33+ denote the power spectral densities of in1, in2, and in3 at the chosen frequency for which the transfer functions H1, H2, and H3 were computed. The power spectral density of the noise signal (Sout+) observed at the circuit's output (node potential v2) is then
The units of the power spectral density are V2/Hz if the observed output signal is a voltage. If the output signal is a current the power spectral density is in A2/Hz.
Sometimes we are want to know how much noise (i.e. what power spectral density) must be injected at the circuit's input to recreate the power spectral density of the noise at the circut's output while assuming all noise sources in the circuit are disabled. This equivalent input noise represents the aggregation of all noise sources at the circuit's input.
To compute it, we need to define the circuit's input (i.e. the independent source that produces the input signal). Suppose for the circuit in Fig. 6 this is uGG(t). First, we need to compute the transfer function from this source to the circuit's output (v2) defined as V2ac/Ugg. We can obtain this transfer function by setting Ugg to 1 while disabling all other independent sources. This produces the following system of equations (which depends on frequency ω).
After solving this system the resulting V2AC corresponds to the transfer function which we denote by H. Because power spectral densities are transformed by multiplying the input power spectral density with the squared absolute value of the transfer function the power spectral density of the equivalent input noise is obtained as
The computation of H and Sout+ needs to be repeated for every frequency for which we would like to compute the power spectral density of the equivalent input noise. If the input signal source is a voltage source the power spectral density of the equivalent input noise is in V2/Hz. If the input signal source is a current source the result is in A2/Hz.When performing noise analysis one has to specify the output signal (which can be any node potential or current appearing in the system of equations as an unknown), the input voltage source or current source (for computing the equivalent input noise), and the frequency range (in the same manner as for AC analysis). The simulator produces two groups of results. The first one comprises the power spectral densities of the output and equivalent input noise along with the output noise contributions of individual noise sources. The second group of results comprises integrals of the computed power spectral densities over the simulated frequency range.