AEiE0606 Digital Signal Processing¶
\(z\)-transform and region of convergence¶
Definition:
\[
X(z)=\sum_{n=-\infty}^{\infty}x[n]z^{-n}.
\]
- ROC is the set of \(z\) values for which the series converges.
- ROC contains no poles.
ROC, causality, and stability¶
For rational systems or sequences:
- Causal right-sided sequence: ROC is outside the outermost pole.
- Anti-causal left-sided sequence: ROC is inside the innermost pole.
- Stable LTI system: ROC includes the unit circle.
Critical exam statement:
- Stability requires the unit circle to lie in the ROC.
- Causality for a rational system means ROC extends outward from the outermost pole.
- A causal system is stable only if the unit circle is also outside all poles and included in the ROC.
Thus, for a causal stable rational system,
\[
\operatorname{ROC}: |z|>r_{\max}, \qquad r_{\max}<1.
\]
“Outside a circle” expresses causality; stability supplies the additional unit-circle condition.
Inverse \(z\)-transform¶
- Methods include power-series expansion, partial fractions, inspection, and contour integral definition.
- The same algebraic expression with different ROCs can correspond to different time sequences.
System function and responses¶
\[
H(z)=\frac{Y(z)}{X(z)}
\]
for zero initial conditions.
- Poles govern natural or transient behavior.
- Zeros shape frequency response and can cancel spectral components.
- Steady-state sinusoidal response is obtained by evaluating on the unit circle if the unit circle is in ROC:
\[
H(e^{j\omega}).
\]
Convolution and Parseval¶
- Time-domain convolution corresponds to multiplication of transforms.
- Parseval relates energy in time domain to energy in transform domain.
DFT and properties¶
The \(N\)-point DFT is:
\[
X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi kn/N}.
\]
Inverse DFT:
\[
x[n]=\frac{1}{N}\sum_{k=0}^{N-1}X[k]e^{j2\pi kn/N}.
\]
Property cues:
- Linearity.
- Periodicity in both time-indexed extension and frequency-indexed repetition.
- Circular time shift corresponds to phase factor.
- Conjugate symmetry for real sequences.
Multiplication of DFTs and circular convolution¶
- Multiplication in the DFT domain corresponds to circular convolution in the time domain.
- Linear convolution equals circular convolution only when proper zero-padding avoids aliasing.
Trap:
- DFT-based fast convolution often requires zero-padding specifically to reproduce linear convolution.
FIR versus IIR¶
| Filter type | Key feature |
|---|---|
| FIR | Finite impulse response, can achieve exact linear phase, usually non-recursive |
| IIR | Infinite impulse response, recursive, efficient for sharp responses but phase usually nonlinear |
IIR design by impulse invariant method¶
- Start from analog prototype.
- Sample the analog impulse response to obtain digital equivalent.
- Preserves impulse-shape relation but can suffer aliasing.
IIR low-pass discrete filter design cue¶
- Often begins with an analog low-pass prototype such as Butterworth or Chebyshev and maps it to digital form.
FIR design by Fourier approximation and window method¶
- Ideal desired frequency response is approximated and truncated.
- Windowing truncates the ideal impulse response and trades main-lobe width, which controls transition width, against side-lobe level, which controls ripple/leakage.
- A window reduces Gibbs oscillation and controls sidelobes; it does not simply reduce transition bandwidth.
| Window | Typical trade-off cue |
|---|---|
| Rectangular | Narrow main lobe, high sidelobes |
| Hamming / Hann | Better sidelobe reduction |
| Blackman | Wider main lobe, lower sidelobes |
DSP examples¶
- If a rational ROC is outside the outermost pole, the sequence is causal.
- If stability is asked, first check whether the unit circle lies in the ROC.
- If DFT-domain multiplication is mentioned, answer circular convolution unless zero-padding converts it to linear convolution.
AEiE0606 revision box¶
- Causal rational ROC: outside outermost pole.
- Stable system: unit circle included in ROC.
- Same poles with different ROC can mean different signals.
- DFT multiplication <-> circular convolution.
- FIR: finite, often linear phase. IIR: recursive, efficient, infinite response.