## 0. Key ideas

DSP #1 was about performing fourier analysis for the continous time signal. Now, time for discrete time signals.

### 1. DTFT

DTFT evaluates discrete-time aperiodic signals.

Compared to CTFT :

• CTFT is for continuous-time aperiodic signal. It has a frequency range $$(-\infty, \infty)$$. Any frequency can exist.

$X(\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega_0t}dt$
• DTFT has frequency range $$2\pi$$. Thus DTFT is periodic, which means that there can only be a limited value of frequency that can exist. However, frequency values are still continous within the $$2\pi$$ range.

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

### 2. z-transform

z-transform is a more general version of DTFT. DTFT don’t always exist/converge, since it cannot be computed for signals that are not stable. However, z-transform can.

Let $z^n$ be an eigenfunction of a discrete-time LTI system. ($$e^{jwn}$$ is that of a continuous-time LTI system)

$x[n] = z^n$ $y[n] = \sum_{k=-\infty}^{\infty}h[k]x[n-k] = H(z)z^n$

$H(z)$ is called a transfer function and is a complex number.

Z-transform is :

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

This equals to DTFT when $$z=e^{j\omega}$$, meaning that DTFT is when z value is on a unit circle in the z-plane. This is also the reason why DTFT is $$2\pi$$ periodic. The benefit of z-transform is that it can be used to evaluate systems that are not stable (ex. feedback loop).

#### Properties of z-transform

$x[k] \Leftrightarrow X(z)$

Time reversal

$x[-k] \Leftrightarrow X(\frac{1}{z})$

Time shift

$x[k-k_0] \Leftrightarrow z^{-k_0}X(z)$

Exponential sequence

$\alpha^k x[k] \Leftrightarrow X(\frac{z}{\alpha})$

z-domain differentiation

$kx[k] \Leftrightarrow -z\frac{d}{dz}X(z)$

#### Region of convergence (ROC)

If $$z=re^{j\omega}$$, then z-tranform looks like DTFT applied to $$x[n]r^{-n}$$.

$X(z) = X(re^{j\omega}) = \sum_{n=-\infty}^{\infty} (x[n]r^{-n})e^{-jwn} = DTFT(x[n]r^{-n})$

Thus, we can say that z-tranform converges if

$\sum_{n=-\infty}^{\infty} \lvert x[n]r^{-n} \rvert < \infty$

If ROC includes the unit circle, we say the system is stable.

#### Poles and zeros

$X(z) = \frac{N(z)}{D(z)}$

$$N(z) = 0$$ are zeros, $$D(z) = \infty$$ are poles. ### 3. DFT

More appropriate name is discrete fourier series. DFT analyzes finite and periodic digital input signal x. The resulting frequency domain is discrete (unlike DTFT).

Compare DTFT with DFT.

$X(\omega) = \sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}$ $X[k] = \sum_{n=0}^{N-1} x[n]e^{-jk\frac{2\pi}{N}n}$

These two are almost equivalent, when $$\omega = \frac{2\pi k}{N}$$. Also, unlike DTFT, input x[n] is finite/periodic.

In other words, we can interprete DFT as DTFT sampled in the frequency domain at $$\omega = \frac{2\pi k}{N}$$ interval.

DFT can be computed via matrix multiplication: A faster algorithm of DFT is FFT.