Dear Soheil,

As the signal is more spread in the time domain it will be more confined in the frequency domain. As an example assume that you have a sine wave which is windowed by a rectangular window, if the width of the window is infinite the frequency domain appears as an impulse with the amplitude of the sine wave. As the window decreases a major lobe and side lobes begin to appear with more spread as the window decreases. This is the consequence of the fact that any signal in time domain can be thought to be a summation of sinusoidal waves with specific frequencies. That is any signal in time domain can be converted to its equivalent representation in frequency domain by a combination of sinusoidal waves. The transform is the Fourier transform.

To demonstrate the effect of the width of the window let us assume that we have a rectangular window with the width T in time domain. This means we have a single rectangular pulse with width T. The frequency domain representation of the pulse will be the well known Sinc x with its known shape of a major lobe and many side lobes with decreasing amplitude as the frequency increases. The main lobe has the extension 2/T and every side lobe has an extension equal 1/T [1]. The main lobe amplitude is equal to the amplitude of the pulse A times its duration T.

So, consequently as T increases the width of the main lobe decreases and its amplitude increases which means that the frequency domain will be more confined in the frequency domain. Th opposite is true.

There will be also the effect of shape of the pulse on the energy contained in the its frequency component. The rectangular pulse will contain strong side lobes because of the abrupt time changes at its edges increases its high frequency content and thereby leads to the growth of the side lobes. If one makes the pulse rectangular then the side lobes will be made to decrease as there is no longer abrupt transitions in time and the high frequency component will be suppressed.

The more smooth the boundary of the pulse, the less will be the high frequency components and the more the side lobes will be attenuated.

Then in summary there are two effects to suppress the side lobes:

Pulse shaping by using smoothing windows

and extending the window width in the time domain.

[1]http://demonstrations.wolfram.com/RectangularPulseAndItsFourierTransform/

Best wishes

Dear Soheil,

As the signal is more spread in the time domain it will be more confined in the frequency domain. As an example assume that you have a sine wave which is windowed by a rectangular window, if the width of the window is infinite the frequency domain appears as an impulse with the amplitude of the sine wave. As the window decreases a major lobe and side lobes begin to appear with more spread as the window decreases. This is the consequence of the fact that any signal in time domain can be thought to be a summation of sinusoidal waves with specific frequencies. That is any signal in time domain can be converted to its equivalent representation in frequency domain by a combination of sinusoidal waves. The transform is the Fourier transform.

To demonstrate the effect of the width of the window let us assume that we have a rectangular window with the width T in time domain. This means we have a single rectangular pulse with width T. The frequency domain representation of the pulse will be the well known Sinc x with its known shape of a major lobe and many side lobes with decreasing amplitude as the frequency increases. The main lobe has the extension 2/T and every side lobe has an extension equal 1/T [1]. The main lobe amplitude is equal to the amplitude of the pulse A times its duration T.

So, consequently as T increases the width of the main lobe decreases and its amplitude increases which means that the frequency domain will be more confined in the frequency domain. Th opposite is true.

There will be also the effect of shape of the pulse on the energy contained in the its frequency component. The rectangular pulse will contain strong side lobes because of the abrupt time changes at its edges increases its high frequency content and thereby leads to the growth of the side lobes. If one makes the pulse rectangular then the side lobes will be made to decrease as there is no longer abrupt transitions in time and the high frequency component will be suppressed.

The more smooth the boundary of the pulse, the less will be the high frequency components and the more the side lobes will be attenuated.

Then in summary there are two effects to suppress the side lobes:

Pulse shaping by using smoothing windows

and extending the window width in the time domain.

[1]http://demonstrations.wolfram.com/RectangularPulseAndItsFourierTransform/

Best wishes

Abdelhalim abdelnaby Zekry

Thanks for your answer.

Conference Paper Method of decrease of discrete Fourier transform sidelobes w...

Maybe it'll help you.

Soheil,

Maybe you could find elements to answer your question by a fine study of the Hann window. As you know, it is usually given by:

h(t)=0.5 - 0.5 cos 2.pi.t/T for t in [0,T]

if you call w(t) the rectangular window on [O,T], you can develop its fourier transform wich is a cardinal sine function. You will note that this Sinc function pass by 0 for f=1/T, 2/T,...k/T. Its first positive side lobe will be between 2/T and 3/T, and the first side lobe (negative valued) is on the interval [1/T, 2/T].

Now looking at h(t), we can write it like:

h(t)=( 0.5 - 0.5 cos 2.pi.t/T ).w(t)

=0.5 [ 1- 0.5 exp(i.2.pi.t/T) - 0.5 exp(-i.2.pi.t/T) ].w(t)

So there is 3 terms to consider for computing the fourier transform of h(t).

1/ the first is mainly the fourier transform of w(t) (divided by 2)

2/ the second corresponds also to the fourier transform of w(t) but the exponentiel part induce a translation of 1/T

3/ the second corresponds also to the fourier transform of w(t) but the exponentiel part induce a translation of -1/T

If you superpose the three terms of this fourier transform you will note that the side lobes are nearly eliminated, except the first (negative valued one) where the three curves combine in a way which will enlarge the first lobes. Hence the situation you describe, and it is more or less the same idea for other windows.

Hope it helps.

Please read my book (The American Universiy Laboratories for Electrical Engineering (part 1))

In it I have a section on this.

Dmitry I Kaplun

Patrice Simard

Ziad Sobih

Thanks for your attention and great helps.

For better reception, detection of the signal and to avoid the interference.