advantage of using complex exponentials in DSP ???

[starfish]

In DSP and signal processing often we use complex exponetial representation of real signals.............what is the basic advantage we are getting from this approach??????

 

[brmadhukar]

For most of the receiver functions complex sampling is mandatory and you have complex numbaers and it makes sense to use complex notation.
B R M

 

[gennar]

The concept of frequency response of the system is defined using Complex exponential because if the input of the system is a complex exponenetial, the output is also a complex exp. with the same freq. but with different amp. and phase.

So, this leads to the concept of freq. response and that the output of the system is the input multiplied by the freq. response.

This is much easier than to use convolution and to say that the o/p is the i/p convolved with the impulse response.

 

[zorro]

The concept of frequency response of the system is defined using Complex exponential because if the input of the system is a complex exponenetial, the output is also a complex exp. with the same freq. but with different amp. and phase.
So, this leads to the concept of freq. response and that the output of the system is the input multiplied by the freq. response.

Right. For any time-invariant linear system, if the input is exp(st), the output is H(s)*exp(st) for any complex s [t from -inf to inf]. The complex H(s) is the transfer function or frequency response. This is the basis of Fourier analysis.
Complex exponentials are the “eigenfunctions” of time-invariant linear systems.

Z

 

[cesare]

Hi All,

complex number is only a convenient mathematical notation. DSP algorithms are however always implemented using conventional operations on real numbers.

 

[cesare]

All the things above are true. However you cannot forget that signals in any case are "real" and all of the DSP algorithms are implemented using real number operatios.

Complex numbers are only a useful matemathical notation.

 

[cedance]

Hi All,

complex number is only a convenient mathematical notation. DSP algorithms are however always implemented using conventional operations on real numbers.

processors might have their own number system such as 1.15 format and there are more numerical implementations.. that is different issue.. but the understanding is very much based on the representation which is more enhanced using the complex exponentials.. as Zorro said.. the input and o/p are very much related.. its very much according to the famous EIGEN VALUES & EIGEN VECTORS!!! if u could know abt it, then u might understand the principle and use of the complex exponentials as to why an i/p with complex exponential and a transfer function gets the result as the multiplication of the two! similar to the Eigen Equation A X = Γ X..... where Γ is the eigen value...

/cedance

 

[spauls]

It will give phase,
as magnitude in one function , that is required by any DSP algo.

 

[konqueror]

i think that it is because complex exponentials make it easy to analyse the linear time invariant systems. if the input is a complex exponential then out put will be also in form of a complex exponential.the frequenccy domain representation can be either in complex exponential or sinusoidal form.it give a good picture of the frequency domain

 

[checkmate]

All the things above are true. However you cannot forget that signals in any case are "real" and all of the DSP algorithms are implemented using real number operatios.

Complex numbers are only a useful matemathical notation.

Actually, I quite agree with cesare. Most dsp design is carried out in the real domain. Signals are always real. So are the filter coefficients. Complex representation only exists in dsp theory.
But complex representation is highly important in EM for impedance matching.

 

[asymbian]

Well,
all the funda of complex nos is basically for mathematical simplicity (or is it?)!
I think the advantage of using exponentials is that instead of getting the frequency components in sines and cosines, we get them as magnitude and phase which are more important from analysis point of view.
The sines and cosines are just brothers out of phase. They are no different... But to understand the signal as a combination of sine and cosine is not that easy. Instead, if you say that this signal is made up of a certain complex exponential (which you have to imagine!) of this MAGNITUDE and this PHASE, it is easier to understand. If you see the Euler's equation, you will understand the subtle conversion from real sines and cosines to complex exponentials...
A lot of imagination is required...

Regards,
asymbian.