Signals & Systems + DSP
Complete formula reference for continuous and discrete-time signals, LTI systems, frequency domain transforms, sampling, and DSP filter design. Includes interactive animations and calculators.
UNIT 1 — Introduction to Signals & Systems
1.1 Signal Classification & Definitions
$ x(t), \quad t \in \mathbb{R} \quad \text{(Continuous-Time)} $
$ x[n], \quad n \in \mathbb{Z} \quad \text{(Discrete-Time)} $
Relationship (sampling):
$ x[n] = x(nT_s) = x\left(\frac{n}{f_s}\right) $
Where $T_s$ = sampling period, $f_s$ = sampling frequency.
1.2 Periodic vs Aperiodic Signals
$ x(t) = x(t + T_0) \quad \forall t \; (CT) $
$ x[n] = x[n + N] \quad \forall n \; (DT) $
DT Sinusoid Periodicity:
$ x[n] = A\cos(\Omega_0 n + \phi) $
Periodic only if $\frac{\Omega_0}{2\pi} = \frac{m}{N}$ is a rational number.
1.3 Energy & Power Signals
$ E_x = \int_{-\infty}^{\infty} |x(t)|^2 \, dt \quad
(CT) $
$ P_x = \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^{T}
|x(t)|^2 \, dt \quad (CT) $
- $E_x < \infty \rightarrow$ Energy Signal ($P_x = 0$)
- $P_x < \infty,>0 \rightarrow$ Power Signal ($E_x = \infty$)
1.4 Even & Odd Decomposition
$ x(t) = x_e(t) + x_o(t) $
$ x_e(t) = \frac{1}{2}\left[x(t) + x(-t)\right] $
$ x_o(t) = \frac{1}{2}\left[x(t) - x(-t)\right] $
1.5 Elementary Signals
$ \delta(t) = 0 \; (t \neq 0), \; \int \delta(t)dt = 1 $
$ u(t) = 1 \; (t>0), \; 0 \; (t<0) $
$ \text{rect}(t/\tau) = u(t+\tau/2) - u(t-\tau/2)
$
$ \text{sinc}(t) = \frac{\sin(\pi t)}{\pi t} $
1.6 Signal Transformations: $y(t) = x(at - b)$
$ x(t - t_0) \rightarrow \text{Time Shift} $
$ x(at) \rightarrow \text{Time Scale} $
$ x(-t) \rightarrow \text{Time Reversal} $
Step 1: Shift first
$\rightarrow x(t - b/a)$
Step 2: Scale $\rightarrow x(at - b)$
UNIT 2 — LTI Systems
2.1 System Properties (Tester)
Linearity
$ ax_1(t)+bx_2(t) \rightarrow ay_1(t)+by_2(t) $
Time Invariance
$ x(t-t_0) \rightarrow y(t-t_0) $
Causality & Stability
$ y(t) \text{ depends only on } \tau \leq t $
$ |x(t)| \leq M_x \Rightarrow |y(t)| \leq M_y $
Interactive Property Tester
2.2 Impulse Response & Convolution
$ y(t) = x(t) * h(t) =
\int_{-\infty}^{\infty} x(\tau)h(t-\tau) \, d\tau $
Properties:
- Commutative: $x * h = h * x$
- Distributive: $x * (h_1 + h_2) = x*h_1 + x*h_2$
- Associative: $x * (h_1 * h_2) = (x*h_1) * h_2$
- Sifting: $x(t) * \delta(t-t_0) = x(t-t_0)$
UNIT 3 — Fourier Series
3.1 Fourier Series Representations
Trigonometric Form:
$ x(t) = a_0 +
\sum_{n=1}^{\infty} [a_n\cos(n\omega_0 t) + b_n\sin(n\omega_0 t)] $
$ a_n =
\frac{2}{T_0}\int_{T_0} x(t)\cos(n\omega_0 t) \, dt $
Complex Exponential Form:
$ x(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0
t} $
$ c_n = \frac{1}{T_0}\int_{T_0} x(t)e^{-jn\omega_0 t}
\, dt $
$ |c_n| = \frac{1}{2}\sqrt{a_n^2 +
b_n^2}, \quad c_n = \frac{a_n - jb_n}{2} $
3.2 Common Signals & Harmonic Synthesis
Square Wave:
$ c_n = \frac{A\tau}{T_0}\text{sinc}(n f_0 \tau) $
Triangular Wave:
$ c_n = \begin{cases}
\frac{4A}{(n\pi)^2} & n \text{ odd} \\ 0 & n \text{ even} \end{cases} $
Sawtooth Wave:
$ c_n = \frac{jA}{2\pi n} \quad (n \neq 0) $
Gibbs Phenomenon & Harmonical Synthesis
UNIT 4 — Continuous-Time Fourier Transform
4.1 CTFT Definition & Common Pairs
$ X(j\omega) =
\int_{-\infty}^{\infty} x(t)e^{-j\omega t} \, dt $
$ \delta(t) \xleftrightarrow{\mathcal{F}} 1 $
$ u(t) \xleftrightarrow{\mathcal{F}} \pi\delta(\omega)
+ \frac{1}{j\omega} $
$ e^{-at}u(t) \xleftrightarrow{\mathcal{F}}
\frac{1}{a+j\omega} $
$ x(t) =
\frac{1}{2\pi}\int_{-\infty}^{\infty} X(j\omega)e^{j\omega t} \, d\omega $
$
\text{rect}\left(\frac{t}{\tau}\right) \xleftrightarrow{\mathcal{F}}
\tau\text{sinc}\left(\frac{\omega\tau}{2\pi}\right) $
$ \cos(\omega_0 t)
\xleftrightarrow{\mathcal{F}} \pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)] $
$ \text{sinc}(Wt)
\xleftrightarrow{\mathcal{F}} \frac{\pi}{W}\text{rect}\left(\frac{\omega}{2W}\right)
$
4.2 Transform Properties & Interactive Explorer
Time Shifting:
$ x(t-t_0) \xleftrightarrow{\mathcal{F}} e^{-j\omega
t_0}X(j\omega) $
Modulation:
$ x(t)\cos(\omega_0 t)
\xleftrightarrow{\mathcal{F}} \frac{1}{2}[X(j(\omega-\omega_0))+... $
Duality:
$ X(t) \xleftrightarrow{\mathcal{F}} 2\pi x(-\omega) $
Convolution:
$ x(t)*y(t) \xleftrightarrow{\mathcal{F}}
X(j\omega)Y(j\omega) $
UNIT 5 — DTFT, DFT & FFT
5.1 Fast Fourier Transform (FFT) Butterfly & Complexity
N-point DFT:
$ X[k] = \sum_{n=0}^{N-1}
x[n]W_N^{kn}, \quad W_N = e^{-j2\pi/N} $
Operations Count:
$ \text{DFT: } O(N^2) $
$ \text{FFT: } O(N\log_2 N) $
$ \text{Speedup} = \frac{N}{\log_2 N} $
UNIT 6 — Laplace Transform & S-Plane
6.1 S-Plane Explorer (Poles & Zeros)
$ X(s) =
\int_{-\infty}^{\infty} x(t)e^{-st} \, dt, \quad s = \sigma + j\omega $
Stability & ROC:
- Causal: ROC is Re(s) > rightmost pole.
- Stable LTI: ROC strictly includes $j\omega$ axis.
- Causal Stable: All poles in Left Half Plane (LHP).
$ \lim_{t\to\infty} x(t) =
\lim_{s\to 0} sX(s) \quad \text{(Final Value)} $
Drag the × poles to observe stability.
UNIT 7 — Z-Transform & Z-Plane
7.1 Z-Plane Explorer (Unit Circle)
$ X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}, \quad z
= re^{j\Omega} $
Stability & ROC:
- Causal: ROC is $|z| > r_{max}$ (outside largest pole).
- Stable LTI: ROC strictly includes Unit Circle $(|z|=1)$.
- Causal Stable: All poles INSIDE unit circle $(|p_k| < 1)$.
$ a^n u[n] \xleftrightarrow{\mathcal{Z}}
\frac{z}{z-a}, \quad |z|>|a| $
Drag pole out of the Unit Circle to break stability.
UNIT 8 — Sampling & Aliasing
8.1 Nyquist-Shannon Theorem & Aliasing Demo
$ f_s \geq 2f_{max} = f_{Nyquist} $
Reconstruction:
$ x(t) =
\sum_{n=-\infty}^{\infty} x[n]\cdot \text{sinc}\left(\frac{t-nT_s}{T_s}\right) $
$ f_{alias} = |f_{signal} - k f_s| $
UNIT 9 — State-Space Representation
9.1 State Equations & Matrix Layout
$ \dot{\mathbf{x}}(t) = A\mathbf{x}(t) +
B\mathbf{u}(t) $
$ \mathbf{y}(t) = C\mathbf{x}(t) + D\mathbf{u}(t) $
Transfer Matrix / Eigenvalues:
$ H(s) = C(sI - A)^{-1}B + D $
$ \det(sI - A) = 0 \quad \text{(Poles)} $
UNIT 10 — DSP Filter Design
10.1 FIR/IIR Design & Window Functions
FIR Window Method:
$ h[n] = h_{ideal}[n] \cdot w[n] $
IIR Bilinear Transform:
$ s = \frac{2}{T_s}\cdot\frac{1-z^{-1}}{1+z^{-1}} $
$ \Omega = 2\arctan\left(\frac{\omega T_s}{2}\right) $