0. Key ideas

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


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.

DFT-49 2

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.

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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.