Visual
Key formulas
| Name | Formula | Variables / Notes |
|---|---|---|
| CTFT | X(jw) = integral from -inf to +inf of x(t) e^(-jwt) dt | X(jw): spectrum; x(t): continuous-time signal; w: angular frequency in rad/s |
| Inverse CTFT | x(t) = (1/2pi) integral from -inf to +inf of X(jw) e^(jwt) dw | x(t): reconstructed signal; X(jw): spectrum; w: angular frequency |
| Laplace Transform | X(s) = integral from 0- to +inf of x(t) e^(-st) dt | X(s): s-domain representation; s = sigma + jw; ROC must include imaginary axis for CTFT to exist |
| Convolution (CT) | y(t) = x(t) * h(t) = integral from -inf to +inf of x(tau) h(t-tau) dtau | y(t): output; x(t): input; h(t): impulse response; tau: dummy variable |
| Z-Transform | X(z) = sum from n=-inf to +inf of x[n] z^(-n) | X(z): z-domain representation; x[n]: discrete-time signal; z: complex variable |
| Parseval's Theorem (CT) | integral |x(t)|^2 dt = (1/2pi) integral |X(jw)|^2 dw | Left side: time-domain energy; Right side: frequency-domain energy; Both must be finite for energy signals |
Key concepts
Energy vs Power signal
An energy signal has finite total energy (E < infinity) and zero average power. A power signal has finite nonzero average power and infinite energy. Periodic signals are power signals. A signal cannot be both.
LTI system properties
A system is LTI if it satisfies both linearity (superposition) and time-invariance (a time shift in input causes the same time shift in output). The output of any LTI system is the convolution of input with the impulse response h(t).
ROC of Laplace Transform
The Region of Convergence is the set of s values for which the integral converges. For causal signals the ROC is a right half-plane. For stable systems the ROC must include the jw axis. Poles are never in the ROC.
Sampling theorem
BIBO stability
A system is bounded-input bounded-output stable if every bounded input produces a bounded output. For LTI systems this requires the impulse response to be absolutely integrable: integral |h(t)| dt < infinity.
Convolution vs Correlation
Convolution computes system output: one signal is flipped before integration. Cross-correlation measures similarity between signals without flipping. Autocorrelation is cross-correlation of a signal with itself.
Tables
Common Fourier Transform Pairs
| Signal x(t) | Transform X(jw) | Notes |
|---|---|---|
| delta(t) | 1 | Flat spectrum |
| 1 | 2pi delta(w) | DC component |
| e^(-at) u(t), a>0 | 1 / (a + jw) | One-sided decaying |
| rect(t/T) | T sinc(wT/2pi) | Bandlimited in frequency |
| cos(w0 t) | pi[delta(w-w0)+delta(w+w0)] | Two impulses |
| e^(jw0 t) | 2pi delta(w - w0) | Single frequency |
Laplace Transform Properties
| Property | Time Domain | s-Domain |
|---|---|---|
| Linearity | a x1(t) + b x2(t) | a X1(s) + b X2(s) |
| Time shift | x(t - t0) | e^(-s t0) X(s) |
| Time scaling | x(at) | (1/|a|) X(s/a) |
| Differentiation | dx/dt | s X(s) - x(0-) |
| Integration | integral x dt | X(s)/s |
| Convolution | x1(t) * x2(t) | X1(s) X2(s) |
System Classification Summary
| Property | Condition (CT) | Test |
|---|---|---|
| Causal | h(t) = 0 for t < 0 | Output depends only on present/past |
| Stable (BIBO) | integral |h(t)| dt < inf | All poles in left half s-plane |
| Memoryless | y(t) depends only on x(t) | No integration or delay |
| Invertible | Distinct inputs give distinct outputs | H(s) has no zeros canceling poles |
Quick facts
- The Fourier transform of a Gaussian is another Gaussian: F{e^(-at^2)} = sqrt(pi/a) e^(-w^2/4a).
- Multiplication in time domain equals convolution in frequency domain, and vice versa.
- A causal LTI system is stable if and only if all poles of H(s) lie strictly in the left half s-plane.
- The DFT of an N-point sequence has N frequency samples spaced 2pi/N apart.
- Time-bandwidth product for a Gaussian is 0.5. No signal can have a smaller product.
- A real-valued signal has a conjugate-symmetric Fourier transform: X(-jw) = X*(jw).
- The Z-transform of a unit step u[n] is z/(z-1) with ROC |z| > 1.
- The initial value theorem: x(0+) = lim as s approaches infinity of s X(s), valid for causal signals.
Exam shortcuts
- Convolution length shortcut: the convolution of a sequence of length M with a sequence of length N produces a sequence of length M + N - 1. For continuous time, if x(t) is nonzero on [a,b] and h(t) on [c,d], then y(t) is nonzero on [a+c, b+d].
- Pole-zero cancellation check: when simplifying H(s), verify no pole-zero pair cancels a pole that would affect stability. A canceled pole in the right half-plane means the system is still unstable, even if the transfer function looks stable.
- ROC shortcut for Z-transform: if the signal is right-sided (causal), ROC is outside the outermost pole. If left-sided, ROC is inside the innermost pole. If two-sided, ROC is a ring between two poles.
- Frequency response from H(s): substitute s = jw directly. The magnitude |H(jw)| is the gain at frequency w. The phase angle of H(jw) is the phase shift. This only works when the ROC includes the jw axis.
- Parseval shortcut for power: for a periodic signal with Fourier series coefficients a_k, average power = sum |a_k|^2. Count only the nonzero coefficients in exam problems to save time.