# When to use Laplace & Fourier Series/Transforms

1. Oct 20, 2007

### MtPiNoY

Hello all,

This is my first post and this seems like an awesome community.

As my first post for the forums, I would like to know the following:

In my classes I learned the Laplace and Fourier Series and Transforms as well as the Z-Transform.

What I do not understand is when do I actually use one over the other? I know the mathematical theory and everything, I just don't know the applications.

Thank you!

2. Oct 20, 2007

### rbj

these are all variations of the same theme. probably most will agree with me that the double-sided Laplace Transform

$$X(s) = \int_{-\infty}^{+\infty} x(t) e^{-st} dt$$

is the most general, and all the other transforms (and Fourier Series is a transform that transforms a single period of a periodic function into an infinite series) derive from Laplace. But pedagogically you might not learn it in that order (and should not).

First, you learn about continuous-time periodic functions and Fourier Series.

Then you extend the concept to nonperiodic functions by letting the period of a continuous-time periodic function increase toward infinity. Then the Fourier Series becomes the Fourier Transform.

The Fourier Transform is enough of a description for decently well defined functions (or "signals") that are not having infinite energy (and, by use of the dirac delta function, can be extended to certain finite power, infinite energy functions, like DC or a sinusoid or a periodic function.)

But for some functions, like the unit-step function, the Fourier Transform gets to be a little icky (doesn't converge nicely), so they change the variable of the transformed function from $\omega$ or $j\omega$ to $s=\sigma+j\omega$ by adding a little real part to $j\omega$ which makes some of these integrals converge.

The Z-Transform, is essentially the Laplace Transform of an ideally sampled signal. it is the counterpart of the Laplace Transform applied to discrete-time signals. (discrete-time signals is the kind of signals that you find in DSP or Digital Control theory. if the signal is being processed by op-amps, resistors, and capicitors, it is continuous-time, not discete-time and the L.T. is what you want. if it's processed by a computer program, then it will likely be uniformly sampled discrete-time and Z.T. is what you want.)

the "Discrete-Time Fourier Transform" (DTFT) is the counterpart of the Fourier Transform for discrete signals.

the "Discrete Fourier Transform" (DFT), of which the so-called "Fast Fourier Transform" (FFT) is a well known implementation technique, is the discrete-time counterpart to Fourier Series. the DFT transforms a periodic and discrete signal in the "time domain" to a periodic and discrete signal in the "frequency domain".

now, in a later post, i'll try to answer your specific question (about what application you use each for), which is a good question to ask. every EE should be asking such and getting it answered satisfactorily. you do not want this to all be some sorta "magic". but you should understand, at least as premises, what i said above, before continuing.

3. Oct 20, 2007

### abdo375

Hey MtPiNoY welcome to physics forum I hope you enjoy it here.

Fourier series has many application in electrical engineering a lot of which fall under signal processing, see the great thing about Fourier series is that it can represent signals as a summation of cosine and sine functions, so if you need to analyze a given system all you have to do is apply a sine or cosine input (and a couple of other unique inputs) to the system and because most signals can be broken down to these two signals then what applies to them applies to more complicated signals, this allows us to analyze the response of complex systems to a great number of inputs without actually applying these inputs.

Laplace Transform: One of the main areas where Laplace transform is essential is circuit theory; as you probably know Laplace can be used to convert differential equations into algebraic ones this allows for analyzing circuits in there transit state without the need to use the regular methods of solving differential equations, this greatly simplifies the effort needed to analyze and solve circuits.

Fourier Transform is problebly the most important transform in electrical engineering because it ties together two of the most used phenomenas known to EE's those of Time and frequency, meaning that if you have a signal or waveform in the time domain and you want to see what frequencies does this signal contain you apply the Fourier transform. looking at the signal in the frequency domain allows us to preform a lot of manipulation on the signal including filtering, sampling, modulation..........etc.

The Z transform is the equivalent of the Fourier transform for the discrete signals.

4. Oct 21, 2007

### MtPiNoY

Thank you for your responses. They were much more in-depth than I had expected, which I appreciate very much. These responses have cleared the clouds for me, and like what you said rbj, I knew the theory and its magic, but am not sure when to apply which to which situations. I am anxious to read your follow up post.

5. Oct 21, 2007

### MtPiNoY

Just to clear something up: is the difference between when you use Fourier Series vs. Fourier Transform solely on whether or not your signal is periodic or non-periodic?

6. Oct 21, 2007

### rbj

oh dear.

i do digital signal processing in audio and music for a living, so this is how I think about it.

nearly always, when i'm thinking about a continuous-time audio signal and it's spectrum, i think in terms of the continuous Fourier Transform that is

$$X(f) = \int_{-\infty}^{\infty} x(t)\ e^{-j 2\pi f t}\ dt$$

i like that one because it is unitary

$$x(t) = \int_{-\infty}^{\infty} X(f)\ e^{+j 2\pi f t}\ df$$

so if i need to think in terms of the Duality Property of the continuous Fourier Transform, it's trivial. no scaling needed. also convolution in either direction has no scalers. (however differentiation and integration and delay operations have an additional $j2 \pi$ factor, but i think it's easier to remember that than to have to remember when to use $1/(2 \pi)$ in convolution.) in reality, signals that we have are in finite duration, so the F.T. can pretty well represent any practical signal.

i used to, when solving a transfer function for an analog filter, routinely substitute $s \rightarrow j \omega$ as a shorthand. didn't mean that i was necessarily expressing it in terms of the Laplace Transform, but it looked like it. sometimes this was handy in doing partial fraction expansion.

but, if doing it in terms of $s$ instead of $j \omega$, then you can talk about (complex valued) poles and zeros of a transfer function. that is normally in the context of the L.T.

What the Laplace Transform is good for, is the time-domain response to some transitioning input (like the step response). Sometimes time-domain behavior is more of interest to you than frequency domain. usually, for audio filters, i think only about the frequency response, but for a system like "portamento" in a mono-phonic music synthesizer, then i am more concerned about the time-domain response. then it's L.T. because you can set up a problem with initial conditions, just as you would set it up to solve with differential equations. but it might be easier to do it with the L.T.

now, pretty much since the early 90's, any actual signal processing that i been writing code for, had to have been Digital Signal Processing, which is nearly always discrete-time signal processing. now here, the thing to remember about is that periodicity in one domain implies discreteness of the other. unformly sampled signals have a spectrum that repeats every multiple of the sampling frequency. so we only need to think about the spectrum from -Nyquist to +Nyquist, which if we normalize goes from $-\pi$ to $+\pi$ which is the principal range in the Discrete-Time Fourier Transform (DTFT). that is really just the Fourier Transform applied to a discrete sequence and it has only terms that repeat every $2\pi$. so you know that it's built-in to sampled data (which is what all we deal with in DSP), it's in its fundamental nature to mirror and repeat at $\pm \pi$. now since it is a discrete sequence, there are no differentiation or integral theorems, but there is[/i] delay expressed as an operation and convolution is there, too.

The Z-transform's relationship to the DTFT is precisely the relationship of the Laplace Transform is to the continuous-time Fourier Transform. So if you were interested in constructing a time-domain response to a transient (like a step response), then doing it with the Z-transform is what you do. otherwise, for me it's just like another shorthand. but instead of

$$s \rightarrow j \omega$$

in the case of connecting the Laplace to Fourier regarding continuous-time signals...

... it's

$$z \rightarrow e^{j \omega}$$

which is what connects Z to DTFT regarding discrete-time signals.

so it really depends on whether you in a continuous-time or discrete-time environment, and what you're trying to do: deal with frequency-domain behavior or deal with time-domain behavior.

7. Oct 22, 2007

### rbj

yeah. i think that's it.

the Fourier Series (i'm gonna express this like an EE, not wussy math guy::grin::) transforms a periodic function in one domain (let's say the "time domain"):

$$x(t) = x(t+T) \ \ \forall t$$

to a discrete sequence of coefficients (of sinusoids) which represents discrete frequencies in the frequency domain:

$$c_n = \frac{1}{T} \int_{t_0}^{t_0+T} x(t) e^{-j 2 \pi (n/T) t} dt$$

where

$$x(t) = \sum_{n=-\infty}^{+\infty} c_n e^{j 2 \pi (n/T) t}$$

so you see that it's periodic (but not necessarily discrete) in one domain and discrete (but not necessarily periodic) in the other domain. that's just like sampled signals (and their spectra) but with the time and frequency roles reversed. the Duality Property of the Fourier Transform lets you just switch time and frequency around (might have to toss in a minus sign).

8. Oct 27, 2007

### user101

rbj, sounds good... what company do you work for?

9. Oct 28, 2007

### rbj

Kurzweil Music Systems, a.k.a. Young Chang R&D .