# Compact summary of Fourier series equations

• rude man
In summary, the author has provided a summary of his thoughts on the Fourier series and its use in sampled data analysis. He believes that the series is the proper way to represent a signal, and that it provides an exact representation of the spectrum of the extended wave including the original signal. He also points out that the Fourier coefficients are equal to the complex DFT coefficients divided by the sampling interval duration.
rude man
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Greg has kindly allowed me to post these equations which I compiled many years ago. Somehow I like them better than anything I've ever run across so maybe someone else will find them useful also.

Actually, I have given some thought to the Fourier series and how they tie in with sampled-data analysis. I believe the series is the proper way to do this. I have a textbook (Stanford prof. W W Harman, Principles of the Statistical Theory of Communication , a believe-you-me not to be scoffed at text, in fact a brilliant one, yet he starts with analyzing a signal x(t) of finite duration T with the footnote that "To make this argument strictly valid, the duration T must be allowed to approach infinity". Without further explanation. He then proceeds to use Fourier transform and time-domain math to synthesize x(t) from the sampled data, which may or may not be exact depending on the existence or non-existence of low-frequency (not high!) components.

But if we think Fourier series we find that we get an exact representation of the signal provided it can be extended to a periodic function of infinite duration, i.e x(t) = x(t+nT) for all integer n.

Note that this precludes for example the existence of a harmonic below ## 2 \pi/T ## since periodicity is then impossible. As an example, if ## x(t) = sin(\omega_1 t) + sin(1.5\omega_1 t) ## then T cannot be the inverse of the lowest harmonic which would be ## 2\pi/\omega_1) ## but must in fact be multiples of ## 6\pi/\omega_1 ##).

The Fourier series thus gives the exact x(t) in terms of the spectrum of the extended wave including the original x(t), 0<t<T. And of course the Fourier coefficients also exactly equal the complex DFT coefficients divided by T.

Well, that's kind of what got me to rewrite those old and decomposing Fourier equations in LaTex. Certainly nothing new or Earth'shattering, just wanted them typed up neatly. Any typos please inform.

## f(t) = \sum_{-\infty}^{\infty} A_n e^{j n \omega_1 t } #### A_n = (1/2 \pi) \int_0^{2 \pi} e^{j n\omega_1 t} d( \omega_1 t) ##We can also say

## f(t) = \sum_{-\infty}^\infty c_n e^{(j \omega_1 n t + \phi_n)} ##
## c_n = |A_n|. c_n ## are real and positive.
## \phi_n = angle ##
## = \tan^{-1} (Im A_n/Re A_n) #### A_{-n} = A_n^* ## always true.Therefore, ## c_{-n} = c_n ##
## \phi_{-n} = - \phi_n ##
Also,

## f(t) = c_0 + 2 \sum_{n=1}^\infty c_n cos(n \omega_1 t + \phi_n ) ##
(NB the cn's are half as big as Skilling's on pp431-432)
Further,

For odd functions [f(-x) = -f(x)]

## A_n ## are purely imaginary
## \phi_n = +/-\pi/2 ##
For even functions [f(-x) = f(x)]

## A_n ## are purely real
## \phi_n ## = 0 or ## \pi ##
For half-wave symmetry [ f(x) = -f(x+T)]

## c_n = A_n = 0 ## for even nNB1: can the sign ambiguity in ##\phi_n ## above be removed a priori?
NB2: ## 2 c_n ## =
peak value of each harmonic n

berkeman

WWGD said:
No, I meant as in :"wrapup" of the relevant equations.
The summation I leave to things like x(t)!

WWGD
rude man said:
No, I meant as in :"wrapup" of the relevant equations.
The summation I leave to things like x(t)!
Got it ( Not so) rude man!

berkeman
WWGD said:
( Not so) rude man!
Don't bet on it! Where there's smoke ...

WWGD
rude man said:
Actually, I have given some thought to the Fourier series and how they tie in with sampled-data analysis. I believe the series is the proper way to do this. I have a textbook (Stanford prof. W W Harman, Principles of the Statistical Theory of Communication , a believe-you-me not to be scoffed at text, in fact a brilliant one, yet he starts with analyzing a signal x(t) of finite duration T with the footnote that "To make this argument strictly valid, the duration T must be allowed to approach infinity". Without further explanation.
What argument, exactly, is being discussed?

The "argument" was the author's prelude to his development of sampling theory leading to the replication of a continuous time function from samples taken from it. I decided to look into the basis of that statement - just curiosity really.

It's common knowledge that DFT resolution and accuracy both depend on sampling interval duration but I wanted to investigate this more quantitatively.

Actually, as I stated , it really started with just my desire to take some old handwritten & faded college-era notes and type them up into LaTex. I was hoping to get hard-copy from them but this goal has eluded me so far.

Perhaps he is referring to this:
Shannon's sampling theorem applies to band-limited waveforms which, by definition, are infinite in extent. It is not obvious a priori how to handle a finite-length waveform that cannot be strictly bandlimited. For a different statement of the problem, note that Shannon's reconstruction kernel (sinx/x) is infinite, so its application to samples of finite support must involve truncation errors and information loss, again making it unclear how to proceed.

A Bell Labs scientist named David Slepian and his colleagues looked at this problem in detail and provided solutions. Their analysis is in terms of prolate spheroidal wavefunctions (PSWF), which are the eigenfunctions of sin x/x. In a series of rather brilliant papers, Slepian et al discussed how to simultaneously band-limit and time-limit a waveform sufficiently in practice and in the presence of noise, and then showed that N=2*W*T samples are sufficient to reconstruct a finite-duration waveform in practice, where W is the bandwidth and T is the duration. This is just the number of samples you get from blindly applying the sampling criterion to a finite-length waveform, but as I say, it is not a trivial result. Slepian finds N by counting degrees of freedom in terms of significant eigenvalues of the PSWF; beyond that number, they fall off dramatically and drastically, indicating that virtually all information is retained in N properly-sampled data values.

Unfortunately, I think his papers are behind a pay wall.

Last edited:
anorlunda and Ibix
His papers in the Bell System Technical Journal used to be widely available online, but recently the journal content was purchased by Wiley and now you have to pay. They are available if you have access to the IEEE Xplore digital library.

EDIT: I found this article of his to be available for free. It's taken from a talk that he gave and is a bit wordy and meandering, but covers some of the material from his technical papers in the Appendix.
Slepian, On Bandwidth

Last edited:
rude man said:
The "argument" was the author's prelude to his development of sampling theory leading to the replication of a continuous time function from samples taken from it. I decided to look into the basis of that statement - just curiosity really.

It's common knowledge that DFT resolution and accuracy both depend on sampling interval duration but I wanted to investigate this more quantitatively.

Actually, as I stated , it really started with just my desire to take some old handwritten & faded college-era notes and type them up into LaTex. I was hoping to get hard-copy from them but this goal has eluded me so far.
A few pages later on in the same Harman text (see attached @ top of p. 34) the author gives a new example of a waveform with 2WT samples taken over 0 to T. Note that in this case he says that (in contrast to previous text) this function is exactly describable by the 2WT samples if samples taken outside the range 0 to T are all zero.

Note that at time t=T the sample is finite. So if an attempt is made to characterize this function in a Fourier series while adhering to the bandwidth limit W we fall flat on our faces. Reason: the discontinuity between the end of one period and the beginning of the next in the extended ( ## -\infty ## to ## + \infty ##) waveform. This discontinuity would violate the limit requirement on W; Fourier coefficients would have to extend to infinite in number.

But note how the requirement of zero samples outside 0<t<T saves the day. Because now it is possible to formulate the extended waveform without discontinuity, e.g by starting and ending in t = ##-\tau ## to t = T+ ## \tau ##. Now there will be Fourier coefficients up to W Hz only. And of course the DFT will also give N (complex) numbers instead of ininite ones since the Fourier coefficients are identical to the DFT computations except for a factor of T.

Can we simply cheat and add two zero samples to the rest? Perhaps, but not if the change results in violation of the W limitation, which probably it would.

So the Fourier series is seen to afford simple yet rich insight into sampling theory and its limitations.

#### Attachments

• Harman, op. cit. p.34 ff.pdf
515.9 KB · Views: 184

## 1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions with different frequencies and amplitudes.

## 2. What are the equations for a Fourier series?

The equations for a Fourier series are:
f(x) = a0 + ∑(ancos(nx) + bnsin(nx)), where n is the frequency and an and bn are the coefficients.

## 3. How is a Fourier series used in science?

A Fourier series is used in science to analyze and represent periodic signals, such as sound waves, electromagnetic waves, and other physical phenomena. It is also used in image processing, data compression, and solving differential equations.

## 4. What is the difference between a Fourier series and a Fourier transform?

A Fourier series is used to represent a periodic function as a sum of sine and cosine functions, while a Fourier transform is used to represent a non-periodic function as a sum of complex exponential functions. Additionally, a Fourier transform operates on continuous signals, while a Fourier series operates on discrete signals.

## 5. What is the importance of the coefficients in a Fourier series?

The coefficients in a Fourier series represent the amplitudes and frequencies of the sine and cosine functions that make up the periodic function. They provide important information about the behavior and characteristics of the original function, and can be used to reconstruct the function from its Fourier series representation.

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