AEiE0605 Signal and system¶
Signal definitions and standard signals¶
A signal is a function of one or more independent variables that conveys information about a phenomenon. The notation \(x(t)\) normally denotes continuous time and \(x[n]\) discrete time.
| Signal | Meaning | Key cue |
|---|---|---|
| Unit step \(u(t)\) | 0 for negative time, 1 for positive time | Turns signals on |
| Delta \(\delta(t)\) | Ideal impulse with unit area | Sifting property |
| Sinc | \(\sin(\pi t)/(\pi t)\) in one common convention | Linked to ideal low-pass behavior |
| Signum | \(\operatorname{sgn}(t)\) | Indicates sign of variable |
Sifting property:
\[
\int_{-\infty}^{\infty}x(t)\delta(t-t_0)dt=x(t_0).
\]
LTI systems, impulse response, and convolution¶
- LTI means linear and time-invariant.
- The impulse response fully characterizes an LTI system.
- Output is convolution of input and impulse response:
\[
y(t)=x(t)*h(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau.
\]
Discrete-time form:
\[
y[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k].
\]
Fourier series¶
- Fourier series represents periodic signals as sums of harmonically related sinusoids or complex exponentials.
- Continuous-time Fourier series applies to continuous-time periodic signals.
- Discrete-time Fourier series applies to discrete-time periodic sequences.
Common property cues:
- Linearity.
- Time shift introduces phase shift.
- Differentiation or multiplication by index changes spectral coefficients in known ways.
- Symmetry in time domain gives recognizable symmetry in coefficients.
Fourier transform¶
- Fourier transform represents aperiodic signals in frequency domain.
- Continuous-time FT pair uses angular frequency or ordinary frequency conventions depending on text.
- Discrete-time Fourier transform is periodic in frequency with period \(2\pi\).
Energy and power spectral densities¶
| Quantity | Used for |
|---|---|
| Energy spectral density | Finite-energy signals |
| Power spectral density | Finite-power random or periodic-like processes |
Parseval-style cues connect time-domain energy or power to frequency-domain integrals or sums.
Power spectral density is average power per unit bandwidth, conventionally measured in W/Hz. Voltage- and current-noise PSDs may instead use V\(^2\)/Hz or A\(^2\)/Hz.
CTFS, DTFS, CTFT, DTFT distinctions¶
| Representation | Signal type |
|---|---|
| CTFS | Continuous-time periodic |
| DTFS | Discrete-time periodic |
| CTFT | Continuous-time aperiodic |
| DTFT | Discrete-time aperiodic sequence |
Common transform properties¶
- Linearity.
- Time shifting corresponds to phase multiplication.
- Frequency shifting corresponds to modulation in time.
- Convolution in time corresponds to multiplication in frequency.
- Multiplication in time corresponds to convolution in frequency.
Signal-and-system examples¶
- If the signal is periodic, think Fourier series before Fourier transform.
- If the system is LTI and input plus impulse response are known, use convolution.
- If a sequence is discrete-time but aperiodic, DTFT is the right spectral representation.
AEiE0605 revision box¶
- LTI output comes from convolution with impulse response.
- Periodic signals use Fourier series; aperiodic signals use Fourier transform.
- DTFT is periodic in frequency.
- CTFS, DTFS, CTFT, and DTFT differ by time domain and periodicity.
- Delta function performs sampling or sifting in integrals.