Zero padding affects the DTFT?

Question

I wanted to understand better how zero padding affects a signal: Which is just N ones. where $ N > 0$ is an Integer $$ X[n] = 1, 1, 1, ... 1 $$ Zero padding it gives: $$ X[n] = 1, 1, 1, ... 1, 0, 0, 0, .. 0 $$ Which is N ones, and K zeros.

I know that Changing the Length of the signal by adding K zeros gives us a different $\omega = \frac{2\pi}{N}$ because N Is different.

When I calculate the Fourier Transform I get:

$\Large \hat X = \frac{1 - e^{-i\omega N}}{ 1 - e^{-i\omega}}$

And I don't understand how it affects the Fourier transform when I plot it on the $\omega $ scale.

Answer 1

The result of zero padding is the same as sampling the DTFT which is a continuous (and periodic) function in the frequency domain. We see this when we compare the formula for the DFT (Discrete Fourier Transform) and the DTFT (Discrete Time Fourier Transform):

DFT:

$$X[k] = \sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}$$

DTFT:

$$X(\omega) = \sum_{n=\infty}^{\infty}x[n]e^{-j \omega n}$$

Note how the DTFT for a waveform of finite non-zero duration is essentially zero padding the DFT to infinity in both directions on the time domain axis, and since time goes to infinity, the frequency domain result will be continuous. So the discrete frequency domain in the DFT with each frequency bin given by $2\pi (k/N)$ becomes a continuous function of frequency as $2\pi (f) = \omega$. If we pad a finite number of samples, we will get a finite number of samples in frequency but the magnitudes will fall on the same result as the DTFT (with a possible phase change based on the linear phase of the implied delay if the zero padding is not symmetric).

In the graphics below, I relate the CTFT (Continuous Time Fourier Transform) to the DTFT and DFT demonstrating some key universal properties with the Fourier Transforms:

Note from above the generalized relationships between time and frequency domains:

• A periodic sequence in time or frequency is only non-zero at discrete samples in the other domain. (example ADC sampling, and the DTFT above).

• A sequence that extends to infinity in time or frequency is continuous in the other domain (CTFT and DTFT above). The Continuous Time Fourier Series Expansion is another example combining both properties, as the time domain waveform is continuous and extends to infinity and is periodic, hence the frequency domain waveform has discrete non-zero values but is continuous as zero in between these values.

• A sequence of finite duration has implied periodicity in the Fourier Transforms based on the cyclical nature of each basis element (example: DFT; the result is mathematically equivalent to repeating the N samples in time to infinity but will result in a continuous frequency domain waveform with zero value between each of the original samples.)