DTFT, DFT and z-transform

Syllabus: Discrete-time Signals: DTFT, DFT, z-transform, discrete-time processing of continuous-time signals.

12 hand-picked visual resources on this topic, 9 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

DFT properties

The DFT is periodic and circular: shifts wrap around, and multiplication in one domain is circular convolution in the other. Applying the DFT four times returns the original scaled by $N^{2}$ .

$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}$
$\text{DFT}\{\text{DFT}\{x[n]\}\}=N\,x[(-n)\bmod N]$
twice = scaled reversal
$X[0]=\sum_n x[n]$
the DC bin is the sum

See it

z-transform and ROC

The algebra is the easy part; the ROC carries the information about causality and stability. Same expression, different ROC, different signal.

$a^{n}u[n]\leftrightarrow\frac{z}{z-a},\ |z|>|a|$

Periodicity of discrete sinusoids

A discrete-time sinusoid is periodic only if $\Omega/2\pi$ is rational. Unlike the continuous case, not every sinusoid repeats.

$\cos(\Omega n)\ \text{periodic}\Leftrightarrow\frac{\Omega}{2\pi}=\frac{k}{N}\ \text{rational}$

DTFT and its properties

The DTFT is periodic in $\omega$ with period $2\pi$ — that periodicity is the frequency-domain picture of sampling, and it is what makes $\omega=\pi$ the highest discrete frequency there is.

$X(e^{j\omega})=\sum_nx[n]e^{-j\omega n},\qquad x[n]=\frac{1}{2\pi}\int_{2\pi}X(e^{j\omega})e^{j\omega n}d\omega$

DFT, circular convolution and the FFT

The DFT samples the DTFT at $N$ points, so everything it does wraps around. Multiplying two DFTs gives circular convolution; getting ordinary linear convolution out of it needs enough zero padding for the true result to fit.

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

Inverting the z-transform, and what the ROC decides

The algebra gives $X(z)$ ; the ROC decides which signal it came from. The same expression is a right-sided, left-sided or two-sided sequence depending on the region, and the question always supplies enough to tell which.

$\text{ROC outside the outermost pole}\Rightarrow\text{causal (right-sided)}$
$\text{ROC inside the innermost pole}\Rightarrow\text{anti-causal (left-sided)}$

Difference equations and the system function

A constant-coefficient difference equation transforms directly into $H(z)$ , with the delays becoming powers of $z^{-1}$ . Reading stability off the poles is then the same exercise as in continuous time, with the unit circle in place of the $j\omega$ axis.

$\sum_ka_ky[n-k]=\sum_kb_kx[n-k]\ \Rightarrow\ H(z)=\frac{\sum b_kz^{-k}}{\sum a_kz^{-k}}$

← Sampling

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