# Differences between laplace transform, z transform and fourier transform

Started by July 14, 2009
```Hi All,

I have studied three diff kinds of transforms, The laplace transform, the
z transform and the fourier transform. As per my understanding the usage of
the above transforms are:
Laplace Transforms are used primarily in continuous signal studies, more
so in realizing the analog circuit equivalent and is widely used in the
study of transient behaviors of systems.

The Z transform is the digital equivalent of a Laplace transform and is
used for steady state analysis and is used to realize the digital circuits
for digital systems.

The Fourier transform is a particular case of z-transform, i.e z-transform
evaluated on a unit circle and is also used in digital signals and is more
so used to in spectrum analysis and calculating the energy density as
Fourier transforms always result in even signals and are used for
calculating the energy of the signal.

Is my understanding correct. What more technical differences exist and
where do all these differences find their application. Would be really
helpful if someone can give an understanding of this and provide links
where i can look up for the same.

Thanks,
Soma

```
```On 14 Jul, 13:58, "somanath17" <somanath....@gmail.com> wrote:
> Hi All,
>
> I have studied three diff kinds of transforms, The laplace transform, the
> z transform and the fourier transform. As per my understanding the usage of
> the above transforms are:
...
> Is my understanding correct. What more technical differences exist and
> where do all these differences find their application. Would be really
> helpful if someone can give an understanding of this and provide links
> where i can look up for the same.

It would be almost impossible to comment on details in your
post without launching a semantics war: Everybody here have
their own views on exactly what is different and what is
similar between these different techniques.

As far as I am concernd, you have seen quite a few main
points:

- The different transforms have more or less the
same purpose
- The different transforms can (but need not) be
related to each others
- The different transforms have different scopes
of use (transient/steady state; discrete/continuous)
- There technicalities are different

As long as you keep that big picture, you will be fine.
You will come to refine your views as you work with
the different transforms on a detailed level, and your
understanding of each one matures.

Rune
```
```1. Start with the Fourier Series for repetitive waveforms, based around sin
& cos.
This is the fundamental theory underlying all the others.

2. Develop this into the exponential form of the Fourier Series.

3. The extend to the Fourier Transform for non-repetitive finite "transient"
waveforms.

4. Then you hit the difficulty that this does not always converge, a
requirement of the
Dirichlet conditions.

5. So you multiply by an arbitrary decaying exponential, e^(-ct), with "c"
never
explicitly defined but with the understanding that it is always big enough
to bring
about convergence.

6.  (5) is essentiallythe Laplace Transform.

7. Some Laplace Transforms do not converge, but all the ones which you will
encounter during your training period will converge.

8. Then you come into the digital world where all hell breaks loose and you
are asked to undertake some flights of fancy which contradict all the maths
that
you will have studied up until now.

9. The train of sampled digital pulses is related to the Unit Impulse, and
you
now need a transform to model Delayed Unit Impulses.

10. The Laplace Transform of a time shift T is e^(sT) where T is the time
duration of
each sample and also the time between samples. Note that T is a fixed
constant and
not the continuous variable t.

11. e^(sT) is messy to write down, and like all mathematicians, we are lazy
and so
we seek a compact way to write it down. We use the substitution Z = e^(sT)
and so
we end up with the Z Transform, but it is really just another way of writing
down Laplace
Transforms which in their turn are a fudged Fourier Transform to bring about
convergence.

12. e^(sT) is of course the exponential form of a complex number (Remember
Lesson 2 of
your complex variable theory?) and can be represented as a vector of unit
magnitude centred
at the origin. Dependent upon the value of T, it traces out a circle which
is where your
Unit Circle (about which you seemed to be confused) came in!

Voila! Zat is Cointreau!

(Quotation from a Brit TV Ad from 20+ years ago)

"somanath17" <somanath.k17@gmail.com> wrote in message
news:u6mdnWwkqdVI78HXnZ2dnUVZ_t6dnZ2d@giganews.com...
> Hi All,
>
> I have studied three diff kinds of transforms, The laplace transform, the
> z transform and the fourier transform. As per my understanding the usage
> of
> the above transforms are:
> Laplace Transforms are used primarily in continuous signal studies, more
> so in realizing the analog circuit equivalent and is widely used in the
> study of transient behaviors of systems.
>
> The Z transform is the digital equivalent of a Laplace transform and is
> used for steady state analysis and is used to realize the digital circuits
> for digital systems.
>
> The Fourier transform is a particular case of z-transform, i.e z-transform
> evaluated on a unit circle and is also used in digital signals and is more
> so used to in spectrum analysis and calculating the energy density as
> Fourier transforms always result in even signals and are used for
> calculating the energy of the signal.
>
> Is my understanding correct. What more technical differences exist and
> where do all these differences find their application. Would be really
> helpful if someone can give an understanding of this and provide links
> where i can look up for the same.

```
```Sorry, I just realised that I went into gabble mode and did not actually
answer the
questions that you put, but I hope it helped your understanding!

"Me" <invalid@invalid.invalid> wrote in message
news:h3iae6\$m5m\$1@news.albasani.net...
> 1. Start with the Fourier Series for repetitive waveforms, based around
> sin & cos.
> This is the fundamental theory underlying all the others.
>
> 2. Develop this into the exponential form of the Fourier Series.
>
> 3. The extend to the Fourier Transform for non-repetitive finite
> "transient" waveforms.
>
> 4. Then you hit the difficulty that this does not always converge, a
> requirement of the
> Dirichlet conditions.
>
> 5. So you multiply by an arbitrary decaying exponential, e^(-ct), with "c"
> never
> explicitly defined but with the understanding that it is always big enough
> to bring
> about convergence.
>
> 6.  (5) is essentiallythe Laplace Transform.
>
> 7. Some Laplace Transforms do not converge, but all the ones which you
> will
> encounter during your training period will converge.
>
> 8. Then you come into the digital world where all hell breaks loose and
> you
> are asked to undertake some flights of fancy which contradict all the
> maths that
> you will have studied up until now.
>
> 9. The train of sampled digital pulses is related to the Unit Impulse, and
> you
> now need a transform to model Delayed Unit Impulses.
>
> 10. The Laplace Transform of a time shift T is e^(sT) where T is the time
> duration of
> each sample and also the time between samples. Note that T is a fixed
> constant and
> not the continuous variable t.
>
> 11. e^(sT) is messy to write down, and like all mathematicians, we are
> lazy and so
> we seek a compact way to write it down. We use the substitution Z = e^(sT)
> and so
> we end up with the Z Transform, but it is really just another way of
> writing down Laplace
> Transforms which in their turn are a fudged Fourier Transform to bring
> about convergence.
>
> 12. e^(sT) is of course the exponential form of a complex number (Remember
> Lesson 2 of
> your complex variable theory?) and can be represented as a vector of unit
> magnitude centred
> at the origin. Dependent upon the value of T, it traces out a circle which
> is where your
> Unit Circle (about which you seemed to be confused) came in!
>
> Voila! Zat is Cointreau!
>
> (Quotation from a Brit TV Ad from 20+ years ago)
>
> "somanath17" <somanath.k17@gmail.com> wrote in message
> news:u6mdnWwkqdVI78HXnZ2dnUVZ_t6dnZ2d@giganews.com...
>> Hi All,
>>
>> I have studied three diff kinds of transforms, The laplace transform, the
>> z transform and the fourier transform. As per my understanding the usage
>> of
>> the above transforms are:
>> Laplace Transforms are used primarily in continuous signal studies, more
>> so in realizing the analog circuit equivalent and is widely used in the
>> study of transient behaviors of systems.
>>
>> The Z transform is the digital equivalent of a Laplace transform and is
>> used for steady state analysis and is used to realize the digital
>> circuits
>> for digital systems.
>>
>> The Fourier transform is a particular case of z-transform, i.e
>> z-transform
>> evaluated on a unit circle and is also used in digital signals and is
>> more
>> so used to in spectrum analysis and calculating the energy density as
>> Fourier transforms always result in even signals and are used for
>> calculating the energy of the signal.
>>
>> Is my understanding correct. What more technical differences exist and
>> where do all these differences find their application. Would be really
>> helpful if someone can give an understanding of this and provide links
>> where i can look up for the same.
>
>

```
```On Tue, 14 Jul 2009 06:58:13 -0500, somanath17 wrote:

> Hi All,
>
> I have studied three diff kinds of transforms, The laplace transform,
> the z transform and the fourier transform. As per my understanding the
> usage of the above transforms are:
> Laplace Transforms are used primarily in continuous signal studies, more
> so in realizing the analog circuit equivalent and is widely used in the
> study of transient behaviors of systems.
>
> The Z transform is the digital equivalent of a Laplace transform and is
> used for steady state analysis and is used to realize the digital
> circuits for digital systems.
>
> The Fourier transform is a particular case of z-transform, i.e
> z-transform evaluated on a unit circle and is also used in digital
> signals and is more so used to in spectrum analysis and calculating the
> energy density as Fourier transforms always result in even signals and
> are used for calculating the energy of the signal.
>
> Is my understanding correct. What more technical differences exist and
> where do all these differences find their application. Would be really
> helpful if someone can give an understanding of this and provide links
> where i can look up for the same.
>
> Thanks,
> Soma

Yes an no.

Laplace transform:  Yes, but it is also widely used for designing control
systems and for determining the stability characteristics of continuous-
time systems.

Z-transform:  The sampled-time (_not_ digital, although it's widely used
in digital systems) version of the Laplace transform.  Useful for all the
same things that the Laplace transform is useful for.

Note that by carefully defining the sampling process you can derive the z
transform from the Laplace transform in a way that is exact.  The usual
derivation defines sampling as multiplication by a series of impulses of
infinite height and finite area; this gives everyone gas pains but is
generally a nice way of thinking of it.  You can avoid impulses at the
expense of convenience (but not rigor, as far as I can tell).

Note also that by carefully defining the reconstruction process you can
derive the Laplace transform from the z transform, if you're so
inspired.  You say ta-mah-toe, I say toe-may-toe.

Your definition of the Fourier transform sounds like it is specifically
referring to the discrete Fourier series (AKA FFT).  Further, your
assertion that it always results in even data is generally incorrect,
although all the flavors of Fourier transform/series will yield even real
output if you present them with even real input.  There are four flavors
of things called Fourier:

The 'real' Fourier transform has a continuous, infinite extent input and
a continuous, infinite extent output.  It is an operation on continuous-
time signals to express them in the frequency domain.  Because the time
variable drops out and a frequency variable comes in, all without losing
any information, it's a transform.

The Fourier series for continuous-time periodic signals has a continuous
finite-time (one cycle) input and a discrete (at the harmonics), infinite-
extent output.  Used for analyzing (what else!) periodic signals.

The Fourier transform for sampled-time signals.  This is _not_ the FFT,
as it takes a discretely sampled infinite extent input and coughs up a
continuous finite extent output.  Like the z transform this can be
derived from the 'real' continuous-time infinite extent Fourier transform.

Properly, the Fourier series for discrete-time periodic signals (whose
period is restricted to an integer number of samples, naturally).  This
is the one that can be turned into the FFT; it takes discrete, finite-
span input data and coughs up discrete, finite-span output data.  Because
the computation is so efficient, the FFT is used extensively to generate
numerical approximations to 'real' Fourier transforms for infinite
extent, continuous-time signals.  This is done by sampling and truncating
the signal (which is where the approximation happens), windowing (which
usually makes the approximation better), performing the FFT, then finally
interpreting the data in a way that makes sense given the sample rate and
length of the sampled signal.

--
www.wescottdesign.com
```
```"Tim Wescott" <tim@seemywebsite.com> wrote in message
news:CcKdnbtkBvUWVsHXnZ2dnUVZ_gJi4p2d@web-ster.com...
> Note that by carefully defining the sampling process you can derive the z
> transform from the Laplace transform in a way that is exact.  The usual
> derivation defines sampling as multiplication by a series of impulses of
> infinite height and finite area; this gives everyone gas pains but is
> generally a nice way of thinking of it.  You can avoid impulses at the
> expense of convenience (but not rigor, as far as I can tell).

There's no need for those "gas pains" ...

1. The transform for the Unit Impulse is derived from a limiting process.

2. A practical limiting process for us is the Nyquist criterion.

3. According to Nyquist, once the sampling pulses have become sufficiently
frequent to ensure that the amplitude-modulated sidebands are not aliasing
with each other, no further advantage (OK, yeah, low pass filtering becomes
easier) is gained by increasing the frequency of the sampling pulses, and,
thereby
reducing their width.

4. To make life easier for ourselves, we like to use a previously calculated
transform
in our analysis, that of the Unit Impulse.

5. However, because of the stopping-short in our limiting process, our
sampling
signal is not a train of Unit Impulses, but a train of T*d(t).

Simple to derive ..... a Unit Impulse is T* (1/T), but our sampling pulses
are
T * 1 ie, 1 volt. So to convert Unit Impulses so that they can represent our
sampling pulses, we must multiply by T

6. Most texts omit this crucial facor of T, the sampling interval, and so
bring about
the "gas pains" .... (Cue: Airry r. been (sp?))

7. Bristow-Johnson has a spiel on the reconstruction process, and finds it
necessary
to introduce this T factor to make things work.

8. If you introduce it at the sampling stage, then all is sweetness and
light.

9. Conclusion. Change all the textbooks to describe sampling as a train of
T*d(t) and
not as a train of d(t), then the gas pains disappear, and peace reigns
throughout the realm.

```
```On Wed, 15 Jul 2009 07:50:46 +0100, Me wrote:

> "Tim Wescott" <tim@seemywebsite.com> wrote in message
> news:CcKdnbtkBvUWVsHXnZ2dnUVZ_gJi4p2d@web-ster.com...
>> Note that by carefully defining the sampling process you can derive the
>> z transform from the Laplace transform in a way that is exact.  The
>> usual derivation defines sampling as multiplication by a series of
>> impulses of infinite height and finite area; this gives everyone gas
>> pains but is generally a nice way of thinking of it.  You can avoid
>> impulses at the expense of convenience (but not rigor, as far as I can
>> tell).
>
> There's no need for those "gas pains" ...
>
> 1. The transform for the Unit Impulse is derived from a limiting
> process.
>
> 2. A practical limiting process for us is the Nyquist criterion.
>
> 3. According to Nyquist, once the sampling pulses have become
> sufficiently frequent to ensure that the amplitude-modulated sidebands
> are not aliasing with each other, no further advantage (OK, yeah, low
> pass filtering becomes easier) is gained by increasing the frequency of
> the sampling pulses, and, thereby
> reducing their width.
>
> 4. To make life easier for ourselves, we like to use a previously
> calculated transform
> in our analysis, that of the Unit Impulse.
>
> 5. However, because of the stopping-short in our limiting process, our
> sampling
> signal is not a train of Unit Impulses, but a train of T*d(t).
>
> Simple to derive ..... a Unit Impulse is T* (1/T), but our sampling
> pulses are
> T * 1 ie, 1 volt. So to convert Unit Impulses so that they can represent
> our sampling pulses, we must multiply by T
>
> 6. Most texts omit this crucial facor of T, the sampling interval, and
> so bring about
> the "gas pains" .... (Cue: Airry r. been (sp?))
>
> 7. Bristow-Johnson has a spiel on the reconstruction process, and finds
> it necessary
> to introduce this T factor to make things work.
>
> 8. If you introduce it at the sampling stage, then all is sweetness and
> light.
>
> 9. Conclusion. Change all the textbooks to describe sampling as a train
> of T*d(t) and
> not as a train of d(t), then the gas pains disappear, and peace reigns
> throughout the realm.

Oooh don't mention That Name -- he may stop taking his meds and get
access to a computer.

I think folks forget just how hard it is to wrap your brain around the
unit impulse once they've found how useful it is and gotten used to it.
I end up explaining this stuff to newbies, and just the notion of a unit
impulse gives people who aren't used to it gas pains.

Just my \$0.02 worth.

--
http://www.wescottdesign.com
```
```On Jul 14, 11:16 am, Tim Wescott <t...@seemywebsite.com> wrote:
.> ......
.> Your definition of the Fourier transform sounds like it is
specifically
.> referring to the discrete Fourier series (AKA FFT).

I would say DFT. I make this perhaps semantic quibble because it is
much easier to explain the properties and applications of the
transform from the definition of the DFT rather than from the
definitions of any of the FFT algorithms that may implement the DFT
efficiently.

.> ...
.> Properly, the Fourier series for discrete-time periodic signals
(whose
.> period is restricted to an integer number of samples, naturally).
This
.> is the one that can be turned into the FFT; it takes discrete,
finite-
.> span input data and coughs up discrete, finite-span output data.

Again just a semantic quibble. This is the DFT which certainly is
usually turner into the FFT. In DSP we select the DFT to implement
because it is the only one of the four transforms that -can- be
calculated, as opposed to symbolically manipulated, because it is
discrete and finite on both sides of the transform. The existence of a
fast algorithm merely makes it computationally accessible without
decades of Moore's Law advances.

.>  Because
.> the computation is so efficient, the FFT is used extensively to
generate
.> numerical approximations to 'real' Fourier transforms for infinite
.> extent, continuous-time signals.  This is done by sampling and
truncating
.> the signal (which is where the approximation happens), windowing
(which
.> usually makes the approximation better), performing the FFT, then
finally
.> interpreting the data in a way that makes sense given the sample
rate and
.> length of the sampled signal.
.>
.Good explanation.

Dale B. Dalrymple
```
```Tim Wescott wrote:

> I think folks forget just how hard it is to wrap your brain
> around the unit impulse once they've found how useful it is and
> gotten used to it.  I end up explaining this stuff to newbies,
> and just the notion of a unit impulse gives people who aren't
> used to it gas pains.

Some temperaments find it harder to accept than others that a
mathematical notion needn't yield to the evolved everyday physics
heuristics called intuition in order to "exist".

Martin

--
Quidquid latine scriptum est, altum videtur.
```
```On Wed, 15 Jul 2009 17:44:37 +0000, Martin Eisenberg wrote:

> Tim Wescott wrote:
>
>> I think folks forget just how hard it is to wrap your brain around the
>> unit impulse once they've found how useful it is and gotten used to it.
>>  I end up explaining this stuff to newbies, and just the notion of a
>> unit impulse gives people who aren't used to it gas pains.
>
> Some temperaments find it harder to accept than others that a
> mathematical notion needn't yield to the evolved everyday physics
> heuristics called intuition in order to "exist".
>
>
> Martin

All I know is that if you fill a room with engineering students and
introduce them to the unit impulse functional, 90% of them will look
confused, and only 10% will look wise.  I was one of the other 10%, and I
can guarantee you that I was _pretending_ to look like I understood until
I could go be confounded in private.

I'm not saying that folk can't appreciate it, or that it isn't
tremendously useful, but you really have to beat your brain into
submission if you want it to be intuitive.

--
www.wescottdesign.com
```