# The phase spectrum of the time impulse function

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#### asymbian

##### Junior Member level 3 Impulse function...

Hi!
I am willing to discuss the phase spectrum of the time impulse function. I am not sure how it looks... Is it "0" or is it a sigma function?

Regards,
asymbian.

#### fletcher

##### Member level 3 Re: Impulse function...

It is '0' for an impulse loacted at t=0. For impulse located at t=t0 it is a constant phase (constant phase shift). (I think so)

#### asymbian

##### Junior Member level 3 Re: Impulse function...

Yes..
Isn't it so that the phase spectrum of a real signal is always an odd function... In case of an impulse at "t0", how will the phase spectrum look? I am trying to visualise how all the waves of all different frequencies from 0 to ∞ "add up" to form the impulse. If all the waves making it up were in phase, the inpulse is generated at a point 0 where the value of all the sines is 0 and cosines is 1. In that case we should get impulses at every such point and not just a single impulse...
I am confused.

Regards,
asymbian.

#### the_jackal

##### Member level 4 Re: Impulse function...

The impulse function or delta function δ(t) is an infinately large amplitude pulse with zero pulse width and unity weight (area under the pulse). In time domain it is represented by a single 'spike' at t=0 and in frequency domain it is constant 1 throughout. There is no phase spectrum for the impulse function!

#### asymbian

##### Junior Member level 3 Re: Impulse function...

Hey!
Every signal has to have an amplitude and phase spectrum! How do we know how the waves are going to add? The phase spectrum being "0" is not the same as not having a phase spectrum at all!

Regards,
asymbian.

#### the_jackal

##### Member level 4 Re: Impulse function...

I meant the phase is zero. What you are probably asking for is a magnitude and phase plot of some signal. As an example you can consider exponential signal.

#### zorro Re: Impulse function...

Hi asymbian,

asymbian said:
Yes..
Isn't it so that the phase spectrum of a real signal is always an odd function... In case of an impulse at "t0", how will the phase spectrum look? I am trying to visualise how all the waves of all different frequencies from 0 to ∞ "add up" to form the impulse. If all the waves making it up were in phase, the inpulse is generated at a point 0 where the value of all the sines is 0 and cosines is 1. In that case we should get impulses at every such point and not just a single impulse...
I am confused.
Imagine a cosinusoidal signal with period T, frequency F=1/T. Add this signal and all its harmonics with equal amplitude. The frequencies are n*F, with n any positive integer. The spectrum is a set of lines at n*F, n from -∞ to ∞ as each cosinusoidal component has a line at n*F and another one at -n*F. (A better analysis would consider complex exponentials).
All the harmonics “add up” only at t=0 and t=m*T (m=any integer), and for other values ot T they tend to cancel to 0. The temporal signal is a “comb” of impulses of infinite height T seconds apart.
Now consider inceasing T (F is decreased). As T goes to Infinity, there is only one impulse at t=0 and the separation between spectral lines becomes null (this separation is F=1/T). You got the coninuous spectrum of the Dirac delta.
The above is mathematically objectionable (incorrect), but i hope it helps to visualize the basic concept.

Regards

Z

points: 2