DFT of sinusoid

Sirs,

Having read on DFT topics, one of the virtues of DFT is supposed to be that
transform of sinusoid is another sinusoid.

Per se, I understand the statement but what I don't understand is that if
we are transforming from time to frequency domain and since sinusoid has a
single frequency then how come transform shows multiple frequencies.

Since it does, what is the meaning of other frequency components that exist
apart from basic frequency in time domain.

Thanms, Manish

On Tuesday, October 16, 2012 7:59:20 AM UTC-4, manishp wrote:
> Sirs,
> 
> 
> 
> Having read on DFT topics, one of the virtues of DFT is supposed to be that
> 
> transform of sinusoid is another sinusoid.
> 
> 
> 
> Per se, I understand the statement but what I don't understand is that if
> 
> we are transforming from time to frequency domain and since sinusoid has a
> 
> single frequency then how come transform shows multiple frequencies.
> 
> 
> 
> Since it does, what is the meaning of other frequency components that exist
> 
> apart from basic frequency in time domain.
> 
> 
> 
> Thanms, Manish

I'm not sure of what you are reading, but the DFT of a sinusoid in not another sinusoid. Assuming you have not applied any windowing other than the inherent rectangular window, the DFT of a sinusoid will either yield a kronecker delta if the sinusoid's frequency is a bin frequency or you will get a function like (N*sin(f))/(N*sin(f)) where N is the transform size. Sometimes this is called a periodic sinc function. When your sinusoid's frequency does not match a bin frequency the measure of the sinusoid's strength leaks across the DFT's bins. Hence this is called leakage. Countless papers have been written about leakage and mitigating it.

IHTH,
Clay

>
>Since it does, what is the meaning of other frequency components that
exist
>apart from basic frequency in time domain.
>
>Thanms, Manish
>

Maybe this thread helps....
http://dsp.stackexchange.com/questions/431/what-is-the-physical-significance-of-negative-frequencies

Le mardi 16 octobre 2012 13:59:20 UTC+2, manishp a �crit�:
> Sirs,
> 
> Having read on DFT topics, one of the virtues of DFT is supposed to be that
> 
> transform of sinusoid is another sinusoid.

Well, either you have read things that are incorrect, or you have not understood them correctly!

As Clay explained, a sinus at Fo in time domain = 1 frequency = 1 pair of symmetrical peaks (-Fo/+Fo) in frequency domain, at least in theory if you assume continuous signals of infinite length.

In practice, leakage can distort things a bit if the frequency is not a multiple of the sampling frequency, or if you don't compute the DFT over an integer number of cycles, as mentioned by Clay.

Can you tell us where you have learned those "virtues" of the DFT ?

Note (don't take it wrong) : Judging by your other posts, I would say you would save a lot of time if you would first read some tutorials about basic mathematical notions such as complex number representations, integration, derivation, dot product, convolution, exponential, Dirac function, ... If you understand these concepts first, signal prcessing will be much easier for you afterwards.

On Tue, 16 Oct 2012 06:59:19 -0500, manishp wrote:

> Sirs,
> 
> Having read on DFT topics, one of the virtues of DFT is supposed to be
> that transform of sinusoid is another sinusoid.
> 
> Per se, I understand the statement but what I don't understand is that
> if we are transforming from time to frequency domain and since sinusoid
> has a single frequency then how come transform shows multiple
> frequencies.
> 
> Since it does, what is the meaning of other frequency components that
> exist apart from basic frequency in time domain.
> 
> Thanms, Manish

What you may be [mis]remembering is: the _derivative_ of a sinusoid (or, 
more generally, and exponential) is another sinusoid/exponential.

>> Thanms, Manish
>
>What you may be [mis]remembering is: the _derivative_ of a sinusoid (or, 
>more generally, and exponential) is another sinusoid/exponential.

Yes. I think I have mixed DFT and system response to a sinusoid input.

I am re-producing the para from the book I have read:

"a sinusoidal input to a system is guaranteed to produce a sinusoidal
output. Only the amplitude and phase of the signal can change; the
frequency and wave shape must remain the same. Sinusoids are the only
waveform that have this useful property. While square and triangular
decompositions are possible, there is no general reason for them to be
useful."

Am 17.10.12 09:30, schrieb manishp:
> "a sinusoidal input to a system is guaranteed to produce a sinusoidal
> output. Only the amplitude and phase of the signal can change;"

Even this is wrong. It only applies to *linear* systems, which is a very 
good approximation to many importing real-world systems for not too 
large amplitudes. For high enough amplitudes, everything gets nonlinear 
eventually. E.g. sound wave propagation is linear for usual amplitudes, 
but becomes nonlinear for the sound of an explosion.

Nonlinearity first manifests itself as higher harmonics. What cannot 
change, is the fundamental frequency of excitation. As long as the 
fundamental changes, you are not in a steady state.

	Christian

On 10/17/12 11:04 AM, Christian Gollwitzer wrote:
> Am 17.10.12 09:30, schrieb manishp:
>> "a sinusoidal input to a system is guaranteed to produce a sinusoidal
>> output. Only the amplitude and phase of the signal can change;"
>
> Even this is wrong. It only applies to *linear* systems,

linear *and* time-invariant.  a time-varying linear system can screw up 
a sinusoid pretty easily.

anyway, even though i think the book as assuming LTI in context (what 
book is it?  what's the title?), i dunno how well it does in explaining 
the significance of sinusoids regarding LTI systems.

"a sinusoidal input to a [linear, time-invariant] system is guaranteed 
to produce a sinusoidal output. Only the amplitude and phase of the 
signal can change; the frequency and wave shape must remain the same."

this is true [with the caveat].  if the LTI is discrete-time, then the 
additional caveat of the sinusoid being below the Nyquist frequency is 
also required for the statement to be true.

"Sinusoids are the only waveform that have this useful property."

this is only sorta-kinda true.  really, it's not true.

"While square and triangular decompositions are possible, there is no 
general reason for them to be useful."

the issue is what are eigenfunctions to LTI systems, and, more 
generally, it is not sinusoids but exponential functions.

an analog, continuous-time LTI is made up of
    1. constant scalers
    2. adders (or subtractors)
    3. integrators (w.r.t. time) or differentiators, if you like.

a digital, discrete-time LTI is made up of
    1. constant scalers
    2. adders (or subtractors)
    3. delays (w.r.t. time) or "negative delays", if you like.

now, IN GENERAL, the class of signals that are eigenfunctions for either 
1., 2., or either 3. are exponential functions with a constant "alpha" 
coefficient in the exponent:

     x(t)  =  A e^(alpha*t)

or

     x[n]  =  A e^(alpha*n)


if those go into an LTI system (or any component, 1., 2, or 3. of an LTI 
system), what comes out is:


     y(t)  =  H(alpha) A e^(alpha*t)

or

     y[n]  =  H(e^alpha) A e^(alpha*n)


where in the continuous-time case, H(s) is the Laplace Transform of h(t) 
and in the discrete-time case, H(z), is the Z Transform of h[n], both 
being the respective impulse responses of the LTI system.  from another 
simple analysis, you can see that the impulse response of an LTI system 
*fully* describes its input/output behavior for *any* signal going in.

but you can see that if you scale the exponential, you get an identical 
exponential, except for the scaling constant (that's what an 
eigenfunction is).  if you add two exponentials with the same "alpha", 
you get a third exponential with that alpha.  that's easy.  that's 
component 1 and 2.

and you can see that if you integrate or delay an exponential, you get 
an identical exponential, but with a scaling constant (that will depend 
somehow on alpha).  that's component 3.  in fact, it is only this 
component 3 that can discriminate between different alphas (and, as 
we'll see, different frequencies).

for sinusoids, that is a special case where alpha is purely imaginary:

    alpha  =  j*omega

and for damped-sinusoids, then alpha is complex:

    alpha  =  sigma  +  j*omega


this is the general class of function that retains its waveshape and 
time-scaling coefficient.  nice, pretty, and real sinusoids are a 
subclass of that.


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On Wed, 17 Oct 2012 17:04:02 +0200, Christian Gollwitzer wrote:

> Am 17.10.12 09:30, schrieb manishp:
>> "a sinusoidal input to a system is guaranteed to produce a sinusoidal
>> output. Only the amplitude and phase of the signal can change;"
> 
> Even this is wrong. It only applies to *linear* systems, which is a very
> good approximation to many importing real-world systems for not too
> large amplitudes. For high enough amplitudes, everything gets nonlinear
> eventually. E.g. sound wave propagation is linear for usual amplitudes,
> but becomes nonlinear for the sound of an explosion.
> 
> Nonlinearity first manifests itself as higher harmonics. What cannot
> change, is the fundamental frequency of excitation. As long as the
> fundamental changes, you are not in a steady state.

It only applies to linear, time-invariant systems.  Mixers in RF work are 
often modeled as multipliers (and often implemented to be as close as 
possible to multipliers).  A functional block that effectively multiplies 
a sine wave by a square wave is linear, but it is time-varying and the 
output isn't a sine wave.

To contradict further (but in a much less general sense), a nonlinear 
system _can_ change the fundamental frequency of excitation, or at least 
significantly attenuate the fundamental and generate totally unrelated 
waveforms.  All you need to do is rectify the input wave, then use that 
to power whatever autonomous oscillator(s) or other signal generators you 
have the power for.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

On Wed, 17 Oct 2012 10:48:05 -0500, Tim Wescott wrote:

> On Wed, 17 Oct 2012 17:04:02 +0200, Christian Gollwitzer wrote:
> 
>> Am 17.10.12 09:30, schrieb manishp:
>>> "a sinusoidal input to a system is guaranteed to produce a sinusoidal
>>> output. Only the amplitude and phase of the signal can change;"
>> 
>> Even this is wrong. It only applies to *linear* systems, which is a
>> very good approximation to many importing real-world systems for not
>> too large amplitudes. 
[snip]

Thanks for filling the missing "linear, time invariant" in.  Meant to 
include that...