Fourier series of a square wave

The Fourier series of a square wave contains only odd harmonics

http://mathworld.wolfram.com/FourierSeriesSquareWave.html

The amplitude of the odd harmonics is 4/(k*pi) for k = 1, 3, 5, ...

If I try to see this in Matlab, I see the amplitude as only 2/(k*pi). Is
there something wrong in my Matlab code (below)?

N = 32768;
fc = 100e3; 
fs = 16*fc;   % sampling frequency
fm = fs/256;  % modulating frequency

x = square(2*pi*fm*(0:N-1)/fs); % square wave varying between +1 and -1

freq = (fs/N)*(-N/2:1:(N-1)/2);
plot(freq/1e3, (fftshift(X));

>The Fourier series of a square wave contains only odd harmonics
>
>http://mathworld.wolfram.com/FourierSeriesSquareWave.html
>
>The amplitude of the odd harmonics is 4/(k*pi) for k = 1, 3, 5, ...
>
>If I try to see this in Matlab, I see the amplitude as only 2/(k*pi). Is
>there something wrong in my Matlab code (below)?
>
>N = 32768;
>fc = 100e3; 
>fs = 16*fc;   % sampling frequency
>fm = fs/256;  % modulating frequency
>
>x = square(2*pi*fm*(0:N-1)/fs); % square wave varying between +1 and -1
 X = fft(x)/N;
>freq = (fs/N)*(-N/2:1:(N-1)/2);
>plot(freq/1e3, (fftshift(X));

Is this the power is split equally in +ve and -ve frequencies?

"sumeer" <sumeerbhatara@yahoo.com> wrote in message 
news:Df2dnQZKJ-qse1DanZ2dnUVZ_uCinZ2d@giganews.com...
> The Fourier series of a square wave contains only odd harmonics
>
> http://mathworld.wolfram.com/FourierSeriesSquareWave.html
>
> The amplitude of the odd harmonics is 4/(k*pi) for k = 1, 3, 5, ...
>
> If I try to see this in Matlab, I see the amplitude as only 2/(k*pi). Is
> there something wrong in my Matlab code (below)?
>
> N = 32768;
> fc = 100e3;
> fs = 16*fc;   % sampling frequency
> fm = fs/256;  % modulating frequency
>
> x = square(2*pi*fm*(0:N-1)/fs); % square wave varying between +1 and -1
>
> freq = (fs/N)*(-N/2:1:(N-1)/2);
> plot(freq/1e3, (fftshift(X));

As far as the code goes, I don't get the plot scaling at all.

I note that you're computing an FFT of a finite sequence.  The Fourier 
Series is taken from -inf to +inf.  So, I'm not surprised they aren't 
exactly the same - but there may be a reason why they would be; I just don't 
know.  The time sequence clearly introduces aliasing.  Also, you have an 
uneven function of time so that the FFT isn't conveniently purely real - 
actually it's purely odd.  Why do that when it might be easier to deal with 
a little bit by making the square wave even in time - and then the FFT 
should be purely real.

Fred

On Mar 4, 4:22 pm, "sumeer" <sumeerbhat...@yahoo.com> wrote:
> >The Fourier series of a square wave contains only odd harmonics
>
> >http://mathworld.wolfram.com/FourierSeriesSquareWave.html
>
> >The amplitude of the odd harmonics is 4/(k*pi) for k = 1, 3, 5, ...
>
> >If I try to see this in Matlab, I see the amplitude as only 2/(k*pi). Is
> >there something wrong in my Matlab code (below)?
>
> >N = 32768;
> >fc = 100e3;
> >fs = 16*fc;   % sampling frequency
> >fm = fs/256;  % modulating frequency
>
> >x = square(2*pi*fm*(0:N-1)/fs); % square wave varying between +1 and -1
>  X = fft(x)/N;
> >freq = (fs/N)*(-N/2:1:(N-1)/2);
> >plot(freq/1e3, (fftshift(X));
>
> Is this the power is split equally in +ve and -ve frequencies?

Real valued signals are represented by complex conjugate
pairs split between the +f and -f bins.  Complex valued
signals are not constrained to split evenly between positive
and negative frequencies in the FT or FFT.

Well,

the problem is that you are trying to bypass too many conceptual steps,
and i explain: the link to MathWorld that you give, examines the "Fourier
Series" of a "continuous-time pulse". However, via the MATLAB code that
you provide, you create a "discrete-time pulse" and in the sequel you take
it's "Discrete Fourier Transform". So, if you want to gain insight on what
happens you will have to examine the discrete-time exactly equivalent
analysis of the link, and then try to simulate it. It is quite possible
that the coefficients of the discrete-time Fourier series are slightly
different.

Manolis

If we talk about the complex Fourier series, Matlab FFT matches with what
it should be

http://cnx.org/content/m0065/latest/

Anyone has any comments?

"sumeer" <sumeerbhatara@yahoo.com> wrote in message 
news:Lt2dnXbcy4FIV1PanZ2dnUVZ_qqgnZ2d@giganews.com...
> If we talk about the complex Fourier series, Matlab FFT matches with what
> it should be
>
> http://cnx.org/content/m0065/latest/
>
> Anyone has any comments?


I don't readily see how you reach that conclusion... Perhaps you can 
illuminate.

----------------

To extend the comments about FFT vs. Fourier Series:

Some identities:

***time <> frequency ... Fourier Transform or as specified - a transform 
pair

***Continuous time, periodic <> Discrete frequency and infinite extent

A Fourier Transform of the continuous, periodic time function will end up 
with discrete frequencies and lends itself to using the Fourier Series 
formulation to get the same thing.

***Discrete time, periodic <> Discrete frequency, periodic

A Fourier Transform of a discrete, periodic time function will end up with 
discrete frequencies AND will be periodic. This differs from a Fourier 
Series of a continuous, periodic time function in that the former is 
periodic while the latter is of infinite extent.

A key difference, no matter how you want to formulate or describe it, goes 
something like this:
1) Start with a square wave with infinitely sharp transitions.
2) Compute the Fourier Series.  It will have infinite extent. This means 
infinite bandwidth.
3) Now, sample the square wave.  This violates the Nyquist criterion because 
the bandwidth of the sampled function isn't limited.  Aliasing in frequency 
occurs.

4) Now, whatever you do with the sampled square wave (e.g. compute a DFT) 
isn't going to match the Fourier Series that you computed in (2) above.

------
To apply the Fourier Series, the time function must be periodic.  Indeed, 
when we compute a Discrete Fourier Transform, that's an assumption that's 
made - since the frequency domain components are perforce discrete and 
harmonically related.

And, if the time waveform is captured for exactly an integer number of 
cycles then this assumption of periodicity in time matches the intended 
waveform of infinite extent.

But, when we sample in time we effectively force the frequency domain to 
become periodic.  And that's not at all similar to the result of computing a 
Fourier Series where the higher frequency components generally decay with 
increasing frequency.

I hope this helps clarify.

Fred

On Mar 5, 9:13 am, "Fred Marshall" <fmarshallx@remove_the_x.acm.org>
wrote:
> "sumeer" <sumeerbhat...@yahoo.com> wrote in message
>
> news:Lt2dnXbcy4FIV1PanZ2dnUVZ_qqgnZ2d@giganews.com...
>
> > If we talk about the complex Fourier series, Matlab FFT matches with what
> > it should be
>
> >http://cnx.org/content/m0065/latest/
>
> > Anyone has any comments?
>
> I don't readily see how you reach that conclusion... Perhaps you can
> illuminate.
>
> ----------------
>
> To extend the comments about FFT vs. Fourier Series:
>
> Some identities:
>
> ***time <> frequency ... Fourier Transform or as specified - a transform
> pair
>
> ***Continuous time, periodic <> Discrete frequency and infinite extent
>
> A Fourier Transform of the continuous, periodic time function will end up
> with discrete frequencies and lends itself to using the Fourier Series
> formulation to get the same thing.
>
> ***Discrete time, periodic <> Discrete frequency, periodic
>
> A Fourier Transform of a discrete, periodic time function will end up with
> discrete frequencies AND will be periodic. This differs from a Fourier
> Series of a continuous, periodic time function in that the former is
> periodic while the latter is of infinite extent.
>
> A key difference, no matter how you want to formulate or describe it, goes
> something like this:
> 1) Start with a square wave with infinitely sharp transitions.
> 2) Compute the Fourier Series.  It will have infinite extent. This means
> infinite bandwidth.
> 3) Now, sample the square wave.  This violates the Nyquist criterion because
> the bandwidth of the sampled function isn't limited.  Aliasing in frequency
> occurs.
>
> 4) Now, whatever you do with the sampled square wave (e.g. compute a DFT)
> isn't going to match the Fourier Series that you computed in (2) above.
>
> ------
> To apply the Fourier Series, the time function must be periodic.  Indeed,
> when we compute a Discrete Fourier Transform, that's an assumption that's
> made - since the frequency domain components are perforce discrete and
> harmonically related.
>
> And, if the time waveform is captured for exactly an integer number of
> cycles then this assumption of periodicity in time matches the intended
> waveform of infinite extent.
>
> But, when we sample in time we effectively force the frequency domain to
> become periodic.  And that's not at all similar to the result of computing a
> Fourier Series where the higher frequency components generally decay with
> increasing frequency.
>
> I hope this helps clarify.
>
> Fred

In other words, don't expect that the interpretations, coefficients
and meanings of an expression with the big funky annotated E will be
the same as those of a similar expression with the tall stretched
annotated S.

YAMV (Your annotations may vary), too.

Dale B. Dalrymple
http://dbdimages.com

"dbd" <dbd@ieee.org> wrote in message 
news:9a8335e8-c446-4665-92e8-992d39b6149b@h11g2000prf.googlegroups.com...
> In other words, don't expect that the interpretations, coefficients
> and meanings of an expression with the big funky annotated E will be
> the same as those of a similar expression with the tall stretched
> annotated S.
>
> YAMV (Your annotations may vary), too.
>
> Dale B. Dalrymple
> http://dbdimages.com

Right.  Except that the big funky annotated E (sigma) will be the same as 
those of a similar expression with the tall stretched annotated S when the 
function to be integrated / summed is discrete.

You can always integrate.  You can't always sum discrete elements unless 
they exist.

Fred 

On Mar 5, 11:34 am, "Fred Marshall" <fmarshallx@remove_the_x.acm.org>
wrote:
> "dbd" <d...@ieee.org> wrote in message
>
> news:9a8335e8-c446-4665-92e8-992d39b6149b@h11g2000prf.googlegroups.com...
>
> > In other words, don't expect that the interpretations, coefficients
> > and meanings of an expression with the big funky annotated E will be
> > the same as those of a similar expression with the tall stretched
> > annotated S.
>
> > YAMV (Your annotations may vary), too.
>
> > Dale B. Dalrymple
> >http://dbdimages.com
>
> Right.  Except that the big funky annotated E (sigma) will be the same as
> those of a similar expression with the tall stretched annotated S when the
> function to be integrated / summed is discrete.
>
> You can always integrate.  You can't always sum discrete elements unless
> they exist.
>
> Fred

What we need to look out for are those times when the tall stretched
annotated S deals with things that are not as they actually exist.

Dale B. Dalrymple