Understanding the discrete time Fourier transformation

[Luk]

Let's consider a signal which is continuous in both time and amplitude. Now we consider the amplitude of this signal at specific time instants only. This is my understanding of sampling a signal in time domain.

When performing a Fourier transform on a time discrete signal, we have to apply the DTFT. The DTFT sums upp all the samples multiplied by a complex exponential function. In this complex exponential function we multiply an angular frequency with the index k. Well, I'm not sure what this is all suppossed to tell me. But I especially can't get my head around the angular frequency, which is somehow normalized on the sampling rate. What does this mean?

 

[Vitro]

Let's consider a signal which is continuous in both time and amplitude. Now we consider the amplitude of this signal at specific time instants only. This is my understanding of sampling a signal in time domain.

When performing a Fourier transform on a time discrete signal, we have to apply the DTFT. The DTFT sums upp all the samples multiplied by a complex exponential function. In this complex exponential function we multiply an angular frequency with the index k. Well, I'm not sure what this is all suppossed to tell me. But I especially can't get my head around the angular frequency, which is somehow normalized on the sampling rate. What does this mean?

Give these videos a try, they might have the answer you need.

https://greatscottgadgets.com/sdr

I watched them quite a while ago and found the concepts very well explained and easy to follow.

 

[sophiecentaur]

But I especially can't get my head around the angular frequency

There is nothing magic about 'angular frequency'. You can describe a signal that's varying sinusoidally in time in terms of frequency f (the number of cycles per second) so
A = A₀ sin(2πft) (or any other time dependent waveform)
OR in terms of the angular frequency, which is the number of radians of phase per second
A=A₀ sin(ωt)
They are totally equivalent but a page full of 2π's is hard to read. So, to repeat, the angle that's referred to in 'angular frequency' is the rate of change (in radians) of phase of each of the Fourier components of the signal. Harmonics still have the same integer ratios for ω as for f.

If you can accept that any signal that's described in the temporal domain (i.e. scope trace) can also be described in the frequency domain. (Ordinary frequencies at this stage). If the waveform is repeated then the Fourier Transform consists of harmonics of the fundamental repeat frequency. If you only have time to analyse a finite length extracted out of the waveform then you have to assume that it repeats itself and you 'loop' the waveform. The only Fourier transform you can do on this signal is called a Discrete Fourier Transform and it introduces errors (artefacts) because of the truncation. You can improve on this by 'windowing' the waveform but that's another matter.

 

[sophiecentaur]

I'm not sure what this is all suppossed to tell me

One way it's possible to look at the FT is that the transform shows how the input time function correlates (that +/-∞ integral) with any chosen frequency. i.e it shows how much of that part of the frequency spectrum is present in the input waveform. The output is a string of values, resulting from that correlation, carried out for all frequencies. For a DFT, the only frequencies with non- zero components can only be harmonics.
You also ask about the relevance to the sampling rate. The rate of the samples (f_s for the calculation will be an integer multiple of the repeat, fundamental, frequency f₀ (2ⁿf in the case of an FFT) so the highest frequency that's recorded with no aliasing will be f_s/2 (Nyquist criterion)