# Power Spectral Density vs FFT: A Noise Analysis Perspective

August 27, 2020

### Key Takeaways

• The FFT of the autocorrelation function of discrete-time noise samples reminds the causality of random stochastic signals.

• Noises of two different frequencies can have the same PDF, but PSD will be different.

• The computation of noise PSD gives the normalized frequency distribution of noise powers and helps the designer modify the circuit design, layout, and architecture for noise reduction.

The frequency-domain analysis illustrates the domination of noise frequencies in the system output.

I remember being a source of ‘noise’ in a music class I attended a few years ago. I was trying to encourage my singing instincts, but every time I tried, I was out of sync with my fellow singers. This is a reminder of how noise can completely ruin a process outcome.  It was indeed an embarrassing experience and I voluntarily quit the music practice. Early identification of noise sources and steps to nullify their effects on the system output is very important in circuit design and analysis. Random noise is capable of degrading the circuit performance and can ultimately tarnish the reputation of the product in the market.

Noise power spectral density (PSD) analysis is a powerful tool to identify the harmonics and electromagnetic emissions in a circuit. PSD indicates the power of noise signals distributed over the frequency. Measuring the noises in the time domain and converting them into the frequency domain is like extracting useful information from bulk amounts of unprocessed data. The characterization of the noise in PSD analysis utilizes the Fast Fourier Transform (FFT) of the autocorrelation function of the discrete noise signal.

The FFT approach is efficient in estimating the spectral information of dominant noise powers. The estimation of noise distributions through PSD analysis helps the hardware engineer adapt the circuit design so that the system works satisfactorily under a specified bandwidth of interest. Let’s explore PSD analysis and its contribution to making our PCBs less vulnerable to noise. We’ll start with the frequency-based classification of noises.

## Overview of Noise Types

Noise in a circuit can be externally or internally generated. The noise gets superimposed into the circuit signals from power sources, ground lines, switching, or other devices. The operating temperature of a system also propagates noise as thermal noise. Based on the frequency spectrum, the noises can be classified as:

1. Wide-band noise: A wide range of amplitudes and frequency components are present in this type of noise.

2. Impulse noise: The sharp spikes that exponentially decay to both sides over a  frequency band. It is intermittent and difficult to detect and analyze.

3. Frequency-specific noise: This type of noise is characterized by constant frequency and the noise amplitude is dependent on the distance of the affected location from the noise source.

Now that we have seen the presence of various frequencies in noise, let's get into the terms that describe the noise.

## Explaining Probability Density Function ❲PDF❳, PSD, and Power Spectrum

At one time or another, it is safe to say that we have all connected voltage probes across our fingers to observe the waveshape on the DSO screen. Each time, the noise signal observed differs in amplitude and shape. This shows the non-deterministic nature of noise and how it is difficult to predict its influence on the system voltages or currents under consideration. But, that doesn’t mean we have to leave the noise unattended.

The noise elimination process starts with the sampling of the random noise signals in the system. A noise amplitude histogram is a good approximation method that gives the likelihood of various noise amplitudes in a system. The probability density function (PDF) is the universal set of all such noise histograms in a system. The area under the PDF is the probability of all possible noise amplitudes and always takes a unity value. From the PDF function, we can calculate the mean, variance, and standard deviation of the noise samples. The PDF estimates the average noise power but nothing related to its frequency distribution. However, our noise reduction mission is concentrated on restraining noise amplitudes of certain frequencies from the measured output quantities.

The PSD expression offers a simplified solution by providing the distribution of signal power of the stationary noise over frequency.  PDF describes the behavior of random noise whereas PSD gives the harmonic information of the noise signals. The PSD, denoted by S❲f❳, of a continuous and discrete-time signal x❲t❳ can be given as:

When we are limiting the PSD to a certain frequency range--ω1 to ω2--it gives the power spectrum for that frequency bandwidth and can be derived from PSD as follows:

The PSD concept is a potential aspect of improving the signal-to-noise ratio (SNR) performance of a circuit.

## Autocorrelation Functions Unfold the Dichotomy of  Power Spectral Density vs FFT

The PSD of a discrete-time noise signal is given by the FFT of its autocorrelation function, R(k)

From the above discussion, we know that PSD gives the noise powers ❲W❳ vs. frequency ❲Hz❳. The sampling of the noise consolidates the noise amplitude occurrences over sufficient time and transforms the analysis from continuous to the discrete-time domain. For discrete-time signals, FFT is the most convenient tool for calculating spectral noise power distribution. However, we go for the FFT of autocorrelation function instead of computing the FFT of the direct discrete-time noise signal. Why!?

Why do we do this? Because, in random stochastic signals such as noise, the past and present values are statistically analyzed to predict future occurrences. The correlation function is a statistical function that investigates the similarity between two random functions.  When we measure the similarity of random discrete-time function x❲n❳ with itself vs x❲n-k❳, the correlation function becomes an autocorrelation function. The autocorrelation function is the best statistical quantity to represent the causality of the stochastic noise signal and hence, find its way into PSD computation. The autocorrelation function R(k) reduces to total average noise power when we measure the similarity of x❲n❳ with itself, i.e, by putting k=0.

## Moving Towards a Noise-Compatible Circuit Design

Noise compliance is one of the benchmarks in the competitive electronics market. For example, the development, design, and manufacturing stages of electronic circuits should comply with the IEC 61000 series standards to diminish EM emission effects. Noise such as thermal noise, fluctuations, EMI, RFI, crosstalks, etc. are unavoidable and they interrupt or distort circuit signals. Negligence or delay in taking noise countermeasures in sensitive control systems can even result in temporary process shutdowns. The general rule in circuit design is to adopt noise amplitude limits and frequency ranges described in product standards set by IEC or CISPR for achieving noise compatible devices.

The tedious task of noise signal identification can be made easier with PSD analysis. Recognizing the noise crowding at a certain frequency or bandwidth from PSD plots puts a full stop to the guesstimation of noise in various circuit networks. The PSD curve plotted on in-rush currents and voltages can segregate the noise frequencies during the transient and steady-state operation of equipment. The placement of capacitors, power sources, and other switching devices in a circuit can induce signal interferences and the PSD plot can help in optimizing the architecture and layout of the board. It can play a crucial part in the development of filtering schemes and anti-noise cancellation (ANC) systems. The PSD plots conserve the signal/power integrity in circuit layouts by sensing the power/ground noises.

Your expectation of a high SNR value in your analog electronic designs or low BER value in integrated circuits is realizable only with noise suppression. The frequency-domain analysis is a classical prerequisite for implementing noise reduction techniques in your circuit.

When you use the right PCB design and analysis software, you’ll have the access to the right circuit analysis features you’ll need to create circuits barely prone to noise. The power-aware solutions in Allegro PCB Designer from Cadence integrate with a full suite of analysis tools for optimizing power and signal integrity in PCB designs. Try this unique toolset when you need to design PCBs for advanced and sensitive applications and you want things done right the first time.