 Numerical Methods

## Fourier analysis of real data sets

Consider a series of $N$ measurements $x_n$ that are made at equally spaced time intervals $\Delta t$. The total time to make the measurement series is $N\Delta t$. A discrete Fourier transform can be used to find a periodic function $x(t)$ with a fundamental period $N\Delta t$ that passes through all of the points. This function can be expressed as a Fourier series in terms of sines and cosines,

$\begin{equation} \label{eq:sines} x(t) = \sum\limits_{n=0}^{n < N/2}\left[c_n\cos\left(\frac{2\pi nt}{N\Delta t}\right)+s_n\sin\left(\frac{2\pi nt}{N\Delta t}\right)\right]. \end{equation}$

Data for $x_n$ can be input in the textbox below. When the 'Calculate Fourier Coefficients' button is pressed, the periodic function $x(t)$ is plotted through the data points. The Fourier coefficients are tabulated and plotted as well. The fft algorithm first checks if the number of data points is a power-of-two. If so, it calculates the discrete Fourier transform using a Cooley-Tukey decimation-in-time radix-2 algorithm. If the number of data points is not a power-of-two, it uses Bluestein's chirp z-transform algorithm. The fft code was taken from Project Nayuki.

$x_n$

$\Delta t =$

 $x$ $t$

Fourier coefficients calculated
from the data above.

$n$  $c_n$  $s_n$
 $c_n$$s_n f ### Power spectrum The power spectrum S_{xx} specifies the power there is in the signal in every frequency interval. This quantity is sometimes called the power spectral density (PSD). $\begin{equation} S_{xx} = c_n^2+s_n^2. \end{equation}$ The notation comes from problems with more variables where S_{xx} is the power spectrum of variable x, S_{yy} is the power spectrum of variable y, and S_{xy} describes the cross correlations of variables x and y. The units of the power spectrum are [x]^2/\text{Hz}, where [x] are the units of the variable x. If the measurement points represent a voltage, the units of S_{xx} are \text{V}^2/\text{Hz}. Sometimes a noise signal is specified as the square root of S_{xx}. A voltage amplifier manufacturer might specify the voltage noise as 10 \text{nV}/\sqrt{\text{Hz}} @ 1 kHz. If they leave the specification @ 1 kHz out, it implies they are using the frequency at which the noise is the lowest. f S_{xx} \sqrt{S_{xx}} For the plot, the \sqrt{S_{xx}} curve was scaled so that the peaks of the two curves have the same amplitude.  S_{xx} f ### Digital filtering A digital filter multiplies the Fourier coefficients by a filter function that can selectively enhance or suppress some frequency components of the signal. Some buttons are provided for filters but it is possible to use any mathematical expression involving f, f_0, and f_1 as the filter function. Filter function: f_0 = f_1 =  a_n$$b_n$ $f$

Filtered fit function $x(t)$

$t$  $x$
 $x$ $t$

Often in an experiment, a preamplifier acts like a digital filter. Usually an amplifier acts like a lowpass filter where the cut-off frequency is lower for higher amplification settings. Sometimes a capacitor is put in series with the amplifier to achieve ac-coupling. In this case the amplifier acts like a bandpass filter.

### Numerical differentiation and integration

The Fourier series can be used to estimate the derivative and the integral of the data series. The derivative is,

$\begin{equation} \frac{dx}{dt} = \sum\limits_{n=1}^{n < N/2}\frac{2\pi n}{N\Delta t}\left[-c_n\sin\left(\frac{2\pi nt}{N\Delta t}\right)+s_n\cos\left(\frac{2\pi nt}{N\Delta t}\right)\right]. \end{equation}$

The high frequency components get amplified by a factor of $f$ when the derivative is taken so it is often useful to filter out the high frequency noise before differentiating. The derivative of the filtered fit is tabulated and plotted below.

$t$  $\frac{dx}{dt}$
 $\frac{dx}{dt}$ $t$

The integral of the Fourier series is,

$\begin{equation} \int\limits_0^{t}x(t')dt' = a_0t+\sum\limits_{n=1}^{n < N/2}\frac{N\Delta t}{2\pi n}\left[c_n\sin\left(\frac{2\pi nt}{N\Delta t}\right)-s_n\left(\cos\left(\frac{2\pi nt}{N\Delta t}\right)-1\right)\right]. \end{equation}$

Here the integration constant was chosen so that the integral is zero at $t=0$. The integral of the filtered fit is tabulated and plotted below.

$t$  $\int\limits_0^{t}xdt'$
 $\int\limits_0^{t}xdt'$ $t$