magnitude and phase for DTFT, etc.

[PG1995]

Hi

Could you please help me with these queries? Thanks a lot.

Regards
PG

 

[steveB]

Q1: You just work out the math, and that's what falls out. Maybe you are having trouble starting it off. To start finding the magnitude of a quotient, remember the rule that the magnitude of a quotient equals the magnitude of the numerator divided by the magnitude of the denominator. In this case the denominator is a real number, so finding the magnitude is easy. The numerator has to be worked out and an obvious trig identity will be needed.

Q2: Again, I'm surprised you don't know how to find the phase (or angle) of a complex number.

 

[PG1995]

Thank you.

Please see If were able to save my face!

Regards
PG

 

[steveB]

Lol, good work PG?

I was confident you already know how to do this. The pressure of exam time and lack of sleep can make us all think less clearly.

 

[PG1995]

and lack of sleep can make us all think less clearly

Well, out of the last 24 hours, I had been in sleeping mode for more than 16 hours!

Best wishes
PG

 

[PG1995]

Hi

Q1: I'm trying to understand the difference between DTFT and DFT in general terms. DTFT is used to find frequency components of a discrete-time continuous signal or sequence and it is periodic over period of 2π, and most importantly it takes into account all the sampled valued from -∞ to +∞. Please correct me if I have it wrong.

On the other hand, DFT takes only a limited number of samples and therefore it's a practically realizable. It gives frequency components for those limited number of samples. Suppose, in a live broadcast many thousands of samples are processed and one needs to know the frequency components contained in those samples for real-time processing and this is in such practical situations DFT comes into play. Is my general understanding of DFT correct? Thanks.

Q2: It is said that FFT is any algorithm used to calculate DFT and there are different algorithms used such as prime-factor, Rader, Bluestein.

Why don't we use the formula directly to calculate DFT instead of some algorthm? I remember that in numerical analysis sometimes one algorithm is preferred over the other because in one case convergence takes place at a fast rate. I also remember from numerical analysis that computers are best at doing iterations and computers require fixed pattern of instruction (i.e. algorithm) to solve problems. Do I have it right? Thanks.

Regards
PG


Helpful links:
1: https://answers.yahoo.com/question/index?qid=20101129112908AArVhts
2: **broken link removed**
3: **broken link removed**
4: https://en.wikipedia.org/wiki/Fast_Fourier_transform
5: https://wiki.answers.com/Q/Difference_between_the_DFT_and_the_FFt?#slide=3

Added 07/20/2019:
1: https://dsp.stackexchange.com/questions/29131/2-pi-periodicity-of-discrete-time-fourier-transform
2: https://dsp.stackexchange.com/quest...ion-of-the-discrete-time-spectrum/29126#29126

 

[steveB]

Q1 The DFT is basically the DTFT of a periodic signal. As you say, in the real world we have limited time sequences and can't deal with nonperiodic signals that extend for all time. If we have a limited time-slice, we can't even know if the signal is periodic or zero for other time values. Hence, we can implement a DFT and just assume the signal is periodic, which leads to discrete frequencies, rather than a continuum, which is also efficient in practice. Then you can decide whether your sequence should be padded with zeros around it before making it periodic, and in this way approximate a nonperiodic signal.

Q2 Do your own test in Matlab. Code up the direct FT formula and compare the speed to any FFT. One is practical and the other is not.

 

[PG1995]

Thank you, Steve.

Q1 The DFT is basically the DTFT of a periodic signal. As you say, in the real world we have limited time sequences and can't deal with nonperiodic signals that extend for all time. If we have a limited time-slice, we can't even know if the signal is periodic or zero for other time values. Hence, we can implement a DFT and just assume the signal is periodic, which leads to discrete frequencies, rather than a continuum, which is also efficient in practice. Then you can decide whether your sequence should be padded with zeros around it before making it periodic, and in this way approximate a nonperiodic signal.

So, DFT is just a DTFT of a 'supposed' periodic signal with a period 'N' where 'N' is number of samples. The frequency spectrum obtained in case of DFT is discrete just like that of Fourier series. But what is the period of frequency spectrum of DFT? DTFT is periodic around 2π. Thank you.

Regards
PG

Helpful links:
1: https://www.dspguide.com/ch10/3.htm

Note to self (added 07/20/2019):
Fourier series spectrum consists of discrete frequencies and these frequencies are not periodic. Likewise, I don't think in case DFT there is any period involved.

 

[steveB]

... But what is the period of frequency spectrum of DFT?

You can choose that. It could be N, or you can pad the time domain signal with zero values and make the series longer. You can look up the Matlab help information for "fft" to get some more details.[/quote]

 

[PG1995]

Thanks.

DFT is just a DTFT of a 'supposed' periodic signal with a period 'N' in time domain where 'N' is number of samples.

The frequency spectrum obtained in case of DFT is discrete just like that of Fourier series. But what is the period of frequency spectrum of DFT? For example, DTFT is periodic around 2π. It says here that the periodicity of DFT in frequency domain can be viewed two different ways.

Q1 The DFT is basically the DTFT of a periodic signal. As you say, in the real world we have limited time sequences and can't deal with nonperiodic signals that extend for all time. If we have a limited time-slice, we can't even know if the signal is periodic or zero for other time values. Hence, we can implement a DFT and just assume the signal is periodic, which leads to discrete frequencies, rather than a continuum, which is also efficient in practice. Then you can decide whether your sequence should be padded with zeros around it before making it periodic, and in this way approximate a nonperiodic signal.

But if you are padding it with zeros and hence approximating a non-periodic signal, then don't you think you will have continuous frequency spectrum rather than discrete spectrum? Thank you.

Regards
PG

Reference:
1: https://www.dspguide.com/ch10/3.htm

 

[steveB]

But if you are padding it with zeros and hence approximating a non-periodic signal, then don't you think you will have continuous frequency spectrum rather than discrete spectrum? Thank you.

You can't pad an infinite number of zeros, so it won't become a continuum. But, in a sense, you can approximate a continuous spectrum by making the frequency resolution very fine.