A.1.1 Fourier Series

Imagine a function $f(t)$ of time with periodicity $T$, the fundamental frequency $f_0=1/T$ and angular frequency ${\omega }_0=2\pi /T$

 (A.1)

This function shall be synthesized by a series of sine and cosine functions, reading

 (A.2)

If (A.2) is integrated over the time interval $t\in [-T/2;T/2]$ and using the orthogonality of the sine and cosine functions, we get for the coefficients

 (A.3)

 (A.4)

 (A.5)

The first coefficient $a_0$ is the mean value of the signal, the others are the cosine ($b_n$) and sine ($c_n$) coefficients . In order to illustrate this harmonic synthesis, we consider a rectangular periodic function

 (A.6)

Obviously the mean is zero $(a_0=0)$, and this function is an odd function $f(t)=-f(-t)$; thus the coefficients $b_n$ corresponding to the even cosine function will vanish. The remaining coefficients $c_n$ read

 (A.7)

In Figure A.1 the function and several Fourier series with an increasing number of coefficients are shown. We see that the function is better and better represented, but due to the infinite slope of the rectangular function, an infinite number of Fourier coefficients would be required for a perfect match.

An entirely equivalent representation can also be formulated by a series of complex Euler functions, this time running from $-\mathrm{\infty }$ to $\mathrm{\infty }$

 (A.8)

The coefficients $a_n$ for $n>0$ can be expressed in terms of the series coefficient from (A.2)

 (A.9a)

 (A.9b)

It is quite illustrative to link these expressions to a harmonic signal with angular frequency ${\omega }_0=2\pi f$

 (A.10)

Thus, for a given frequency $n\omega$ we get with (A.8)

 (A.11)

An alternative and instructive derivation of this equation can be found by applying an integration over the period for (A.8) multiplied by $e^{-jm{\omega }_{0}t}$, leading to

 (A.12)

If we integrate (A.12) over the period $[-T/2;T/2]$, we get

 (A.13)

We see that the right hand side is only non-zero for $m=n$. This is called the orthogonality property of the $e$-function in the integration, hence

 (A.14)

This shows in an instructive way how the orthogonality relation is used to derive an alternative expression for $a_n$ to (A.9a)-(A.9b)

 (A.15)

If we replace $e^x=\cos x+j\sin x$ the former Equations (A.2) and (A.8) as far as the related coefficients $a_0$,$b_n$ and $c_n$ can also be derived. The squared value of the periodic function would have the following consequence for Fourier series

 (A.16)

With (A.8) we get

 (A.17)

Performing the integration and using the orthogonality relationship giving zero for $m\ne n$ we get and the total squared signal power is equal to the sum of the square of all coefficients.

 (A.18)

A.1.2 Fourier Transformation

For stationary periodic processes the Fourier series is the perfect tool to investigate the harmonic content of a signal. The frequency resolution – the difference between two frequencies – is restricted by $\mathrm{\Delta }\omega ={\omega }_0=\frac{2\pi }{T}$ because of the periodicity.

In practical applications many nonperiodic or transient signals occur. For the frequency analysis of such signals we require a finer resolution and a different approach. This can be achieved if we perform the limit process for $T$ to infinity for the Fourier series

 (A.19)

Now we let $\mathrm{\Delta }\omega \to 0$. Arranging (A.19) in such a way that we eliminate all $T$ by $\frac{2\pi }{\mathrm{\Delta }\omega }$ and ${\omega }_0$ by $\mathrm{\Delta }\omega$

 (A.20)

 (A.21)

where we have also used $\omega =n{\omega }_0$. Finally, we get the expression for the pair of Fourier transforms by performing the limit process $T\to \mathrm{\infty }$ and $\mathrm{\Delta }\omega \to 0$

 (A.22a)

 (A.22b)

Obviously, the Fourier transform (FT) of a harmonic signal does not converge because the infinite integration requires $f(t)\to 0$ for large arguments. The Fourier transform is a mathematical tool to investigate transient or pulse shaped signals. In order to illustrate some applications of the Fourier transform we investigate a function of rectangular shape

 (A.23)

with the following Fourier transform

 (A.24)

A.1.3 Dirac Delta Function

Based on this rectangular pulse the $\delta$-function is derived, which is a very powerful tool for the description of mechanical and acoustical systems. We use the rectangular pulse with slightly adjusted parameters

 (A.25)

The delta function is not a function in the classical sense, because is has infinite values $A$ at $t=0$ due to $\tau \to 0$. But, it is very useful due to the properties that are presented next. If we integrate the delta function with the above derivation from the rectangular pulse, we find

 (A.26)

If the delta function is centred at $t_0$, the same properties read

 (A.27)

The use of the delta function is given by its so-called sifting property that extracts any function if it occurs in the integral over a product with the delta function, such as

 (A.28)

The first term of the above equation is called the convolution of $f(t)$ and the $\delta$-function. Using this the Fourier transform of the delta function reads as

 (A.29)

Thus, the frequency content of the delta function extends from negative to positive infinity with a constant value. On the other hand the inverse Fourier transform of the exponential function gives

 (A.30)

Exchanging $\omega$ by ${\omega }_0$ and $-t$ by $t_0$ provides the FT of the exponential function.

 (A.32)

A.1.4 Signal Power

The signal power in the time domain is related to the FT similar to (A.18). With

 (A.33)

and change of the integration order we get

 (A.34)

Using (A.30) enables us to replace the parentheses by $2\pi \delta ({\omega }_1-{\omega }_2)$, and then using the sifting property (A.28) gives

 (A.35)

This is known as Parseval’s formula. The total energy in the signal is associated with the integral over all frequency components. If we switch back to the frequency $f$ and $d\omega =2\pi df$ we get rid of the normalization by $1/2\pi$

 (A.36)

A.1.5 Fourier Transform of Real Harmonic Signals

Even if the FT of harmonic signals will not provide a finite result, the delta function enables us to apply the FT even to harmonic signals. The cosine function is linked to the exponential function by

 (A.37)

Using (A.32) we get for the FT of the cosine function

 (A.38)

Applying the infinite integration interval of the FT to the cosine function leads consequently to infinity spectra in the frequency domain, here given by two symmetric delta functions with peaks at $\omega =\pm {\omega }_0$.

A.1.6 Useful Properties of the Fourier Transform

Working with Fourier transforms is much easier when specific relationships are used that link the time domain with the frequency domain. We start with the partial derivative with regard to time.

 (A.39)

Thus, the derivative in time domain corresponds to the multiplication by a factor of $j\omega$ in the frequency domain. This is the same factor that we would get if the time derivative is taken from harmonic signals (1.26) with time factor $e^{j\omega t}$. Both conventions are in agreement when we choose $e^{j\omega t}$ for the time dependence of the harmonic function and $e^{-j\omega t}$ for the exponential function in the Fourier transform.

Time reversal in the time domain means complex conjugate in frequency domain.

 (A.40)

 (A.41)

A further important theorem is the shift theorem which states

 (A.42)

This can be easily proven by exchanging variables. A time delay $\tau$ creates a specific phase change for every frequency.

A very important relationship results from the FT of the convolution of two time signals. This is the infinite integral over the product with time delay $\tau$ for one function

 (A.43)

The Fourier transform of the convolution gives:

 (A.44)

or

 (A.45)

Thus, the Fourier transform of the convolution of two signals is the product of the Fourier transform of each function. It can be further shown that the inverse Fourier transform of a product of spectra is also a convolution in frequency domain.

 (A.46)

A.1.7 Fourier Transformation in Space

The Fourier presentation of time signals can be converted to the wave forms in space. We apply the pair of Fourier transformations from (A.22a) and (A.22b) to the space domain

 (A.47a)

 (A.47b)

This can be further extended to multidimensional applications, for example two dimensional surfaces

 (A.48a)

 (A.48b)

A.2.1 Fourier Transform of Discrete Signals

For dealing with this phenomena, a mathematical formulation of the sampling process is required. The sampling can be represented by multiplying the continuous signal by a sum of delta functions

 (A.49)

called a delta comb. Here, $\mathrm{\Delta }T$ is the sampling period. The rectangular bracket denotes the discrete argument $n$ here. In order to derive the effect of this sampling process to the spectrum we apply the Fourier transformation (A.22a) to (A.49)

 (A.50)

and get with the sifting property

 (A.51)

In Figure A.3 the results of Equation (A.49) are presented graphically. The continuous function is now represented by an infinite train of delta functions. The area of each delta peak at $n\mathrm{\Delta }T$ equals the function value at this time according to Equation (A.28), and it is denoted by the length of the arrow.¹

The sampling has a very peculiar effect on the Fourier transform. Note that the discrete FT has an argument of the form $e^{j\omega \mathrm{\Delta }T}$. In order to understand the effect of sampling in the frequency domain, we interpret the delta comb as a periodic signal with period $\mathrm{\Delta }T$ and the fundamental sampling frequency ${\omega }_s=\frac{2\pi }{\mathrm{\Delta }T}$. The Fourier series in exponential form (A.8) is applied to the delta comb, giving

 (A.52)

In the Equation (A.15) for the complex Fourier coefficients $a_m$, there is only one delta peak in the integration interval.

 (A.53)

With these coefficients the Fourier series of the delta comb is given by

 (A.54)

Entering this into (A.50)

 (A.55)

as far as rearranging sum and integration of the Fourier spectrum of the sampled signal leads to the following expression

 (A.56)

This term in parentheses is the Fourier transform with frequency argument $\omega -m{\omega }_s$ in the exponential function, hence

 (A.57)

Thus, the sampling of the function $f(t)$ leads to a periodic repetition of the spectrum $\mathit{F}(\omega )$ at $n{\omega }_s$ with $n\in \mathbb{Z}$. This effect is illustrated in Figure A.4. The periodic summation of the original spectrum requires a band limited spectrum. If the original spectrum has contributions above ${\omega }_s/2$ or below $-{\omega }_s/2$ there will be an overlap. When sampling continuous signals the sampling rate must be at least twice the maximum frequency content of the analog signal to avoid this effect called aliasing. In signal acquisition systems low pass filters make sure that there is no spectral content in the critical range.

In Figure A.5 the spectrum of the sampled signal is shown when the analog signal has frequency contributions above the maximum frequency ${\omega }_s/2$.

We can conclude on the first question: What is the effect of sampling? The sampling process gives a discrete set of data values approximating the continuous signal. The frequency domain of the original spectrum is converted into a periodic spectrum with frequency interval ${\omega }_s=2\pi /\mathrm{\Delta }T$.

A.2.2 The Discrete Fourier Transform

As digital signals will not contain an unlimited number of samples, the infinite sum has to be replaced by a sum over $N$ samples, and we get the discrete Fourier transform (DFT).

 (A.58)

From this formula all continuous values of $\omega$ can be calculated. But, there is evidence that a spectrum derived from $N$ samples cannot contain more information as is given by these $N$ samples. So, we have to select $N$ points in the spectrum. A natural choice for this is to select $N$ samples in the spectrum between $0$ and ${\omega }_s$

 (A.59)

and the discrete Fourier transform reads

 (A.60)

It can be proven (see e.g. Oppenheim et al., 1999) that we can exactly recover the samples $f[n]$ by

 (A.61)

To conclude, the finite number of samples (and not the sampling procedure) causes a finite number of samples in the Discrete Fourier transform. In numerical tools the above formulas of the DFT are implemented in a numerically more efficient way: the Fast Fourier transform (FFT). A special algorithm is implemented that recursively calculates the DFT but only for $N=2^M$. Newer developments allow for arbitrary lengths by using the FFTW algorithm (Frigo and Johnson, 2005). We don’t care about the details, because most signal analysis packages in Matlab or NumPy provide methods for calculating the FFT efficiently.

A.2.3 Windowing

In principle we have all the means to investigate time signals and their spectral content numerically. There is one remaining issue that is worth mentioning. The spectrum is sampled at discrete values. What happens when we analyze a harmonic signal of frequency $\omega$ that is, for example, between two sampling points ${\omega }_k$ and ${\omega }_{k+1}$?

In order to illustrate this, we switch back to continuous signals. The finite number of samples of the DFT can be interpreted as a rectangular window function. So we multiply the time signal with a function $w(t)$.

 (A.62)

According to (A.46) this results in a convolution in frequency domain

 (A.63)

In other words, the spectrum of a windowed function is the convolution of the FT with the FT of the window function. The FT of a rectangular window that is defined as follows

 (A.64)

has the FT in the form of a sinc function

 (A.65)

Consider now the spectrum of a cosine function (A.38) that consists of two symmetric peaks. When convoluted with the FT of a window function, the spectrum looks as shown in Figure A.6 where the FT of the window appears at the positions of the delta function. The peak of the window function is given by

 (A.66)

being $T_w$ for the rectangular window. So, the amplitude of the cosine function can be derived from the peak value if the maximum is divided by $W(0)/\pi$.

In Figure A.6b the shape of the windows FT is shown, and the sampling of the related discrete FT is denoted by blue dots. The sampling occurs exactly at the zeros of the sinc function. Thus, when the frequencies of the cosine function fit exactly to one spectral sampling value, you get a single peak. What happens if this is not the case? In this case the sampling occurs at the sides of the window peak that is very steep so that the amplitude is underestimated. This is called spectral leakage.

For avoiding this there are two options:

1. We increase the spectral sampling rate by extending the time interval artificially by adding zero values to the time signals, the so-called zero-padding.
2. We use a window function with a broader peak in the frequency domain to mitigate the leakage effect.

An often used window function is the Hanning window given by

 (A.67)

The effect of this broader window is sketched in Figure A.7. The leakage is reduced, but a certain underestimation of the amplitude is still possible.

Mitigating spectral leakage requires compromises; either you need fine spectral resolution, but then you accept high leakage. When choosing a broad peak you reduce the leakage but you loose spectral resolution. A further option is to take a higher number of samples or increase the time length $T_w$ of the window. However, this is not always possible.

The sample interval in the spectrum is $\mathrm{\Delta }\omega =\frac{2\pi }{N\mathrm{\Delta }T}=\frac{2\pi }{T_w}$, and this doesn’t depend on the sampling rate of the time signal but on the size of the time window $T_w=N\mathrm{\Delta }T$. The highest considerable frequency sample is ${\omega }_s/2=\pi /\mathrm{\Delta }T$. The sampling rate influences the width of the spectrum or the maximum frequency value that can be considered.