DFT Leakage

> Hi,
> 
> When the DFT of an integer multiples of the input signal is taken,
> then the
> mahgnitude response shows that there are no sidelobes.
> 
> However, when the DFT of a non-interger multiples of the input signal
> is taken,
> we get the sidelobes, which we call the DFT leakage.
> 
> Ex: 
> when input signal has 5 cycles, and you obtain the DFT at any analysis
> frequency(m*fs/N), we get only the main lobe.
> 
> But, when input signal has 5.6 cycles, along with the main lobe, we
> also get the
> side lobes.
> 
> So, how does the difference of 0.6 cycles in the input signal cause
> the side lobes?

It doesn't. It's the first situation that causes the discrete Fourier 
coefficients of the finite-length sinc, to coincide with the zeros
of the sinc. If you interpolate the spectrum (e.g. by zero-padding)
you will see that the side lobes are there, only between the original 
coefficients:

%%%%%%%%%%%%%  Begin Matlab Script %%%%%%%%%%%%%%%%%%%%%%%%

N= 16;                        % Number of samples in signal
fs=1;                         % Sampling frequency
f= fs/4;                      % Frequency of sinusoidal

tv=[0:N-1];                   % Time vector
s=sin(2*pi*tv*f);             % Signal

S=abs(fft(s));                % Discrete spectrum
fv=[0:N-1]/(N)*fs;            % Frequency vector

Nfft=1024;                    
fvzp=[0:Nfft-1]/Nfft*fs;      % 'Continuous' spectrum
S2=abs(fft(s,Nfft));        

plot(fv,S,'r*',fvzp,S2,'b')   % Plot spectra

% Bring figure to the foreground
figure (gcf)

%%%%%%%%%%%%%  End Matlab Script %%%%%%%%%%%%%%%%%%%%%%%%%

I have this vague recollection that such issues were briefly touched
upon in a recent thread of yours...? Perhaps you should at least try 
to contemplate the answers you get before you ask again?

Rune

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message 
news:f56893ae.0409300213.5fdf85d0@posting.google.com...
>> Hi,
>>
>> When the DFT of an integer multiples of the input signal is taken,
>> then the
>> mahgnitude response shows that there are no sidelobes.
>>
>> However, when the DFT of a non-interger multiples of the input signal
>> is taken,
>> we get the sidelobes, which we call the DFT leakage.
>>
>> Ex:
>> when input signal has 5 cycles, and you obtain the DFT at any analysis
>> frequency(m*fs/N), we get only the main lobe.
>>
>> But, when input signal has 5.6 cycles, along with the main lobe, we
>> also get the
>> side lobes.
>>
>> So, how does the difference of 0.6 cycles in the input signal cause
>> the side lobes?
>
> It doesn't. It's the first situation that causes the discrete Fourier
> coefficients of the finite-length sinc, to coincide with the zeros
> of the sinc. If you interpolate the spectrum (e.g. by zero-padding)
> you will see that the side lobes are there, only between the original
> coefficients:

Sandeep,

This relates very much to your question regarding the interpretation of the 
DFT.  Remember there are two interpretations:

1)  The DFT is samples of the DTFT.  In this case you have windowed the 
signal with a rectangle/boxcar function and hence you are convolving with a 
sinc in frequency.  As Rune mentioned it just so happens that your samples 
are falling at the zeros of the sinc and you don't see them.  Note that if 
you actually use a Hamming window or some other kind of window you can make 
the smearing not so apparent for the case where you're not sampling at an 
even multiple of your tone.

2)  The DFT represents the exact frequency content of a periodic signal. 
For the case where the sampling frequency is a multiple of your tone, this 
is exactly the case.  You have a periodic input and you give exactly one 
period of info to the DFT and what you get out is the EXACT frequency 
content of your infinite tone.  As Stephan and Glen mentioned, for the 
non-integer multiple sampling you are introducing a discontinuity in what 
the DFT then sees as the infinite signal and hence introducing a bunch more 
frequency content.  If you think about the effects of a Hamming window in 
this context, what you are doing when multiplying by a Hamming window is 
tapering both sides of your DFT input sequence to zero.  That way you are 
eliminating the discontinuity and lessening the artifacts.

As you can see both interpretations explain this "mysterious" behavior in 
the DFT.  For this particular case both interpretations work well.  As you 
get into more complicated signals you will find that interpretation 1 is 
typically a more useful way to think about things since you will not be 
analyzing a periodic signal.  Still, interpretation 2  has its uses so it's 
nice to know.

I think you've hit some of the core issues of DSP so take some significant 
time to deeply understand them.  Play around with the equations, experiment 
in Matlab, perhaps read a few different sources to see how they explain it. 
When I was in school the Oppenheim and Schaeffer book was my bible though 
I've heard great things about the book by Proakis and Manolakis as well as 
the book by Rick Lyons.

Brad

allnor@tele.ntnu.no (Rune Allnor) wrote:

>I have this vague recollection that such issues were briefly touched
>upon in a recent thread of yours...? Perhaps you should at least try 
>to contemplate the answers you get before you ask again?

Well, he did manage to catch onto the term "DFT Leakage".  :-)

Robert

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