Zak Transform Yes! There is a transform called the Zak Transform , though it’s much less commonly mentioned than the Fourier or Z-transform. It comes up mostly in signal processing, time-frequency analysis, and communications . 1. Definition The Zak Transform of a signal x(t) (continuous-time) is defined as: $$ Z_x(t, f) = \sum_{n=-\infty}^{\infty} x(t - nT) e^{j 2 \pi n f T} $$ Where: x(t) is the input signal T is a fixed period t is a continuous variable within [0, T) f is the frequency variable It essentially maps a 1D signal into a 2D representation in time-frequency space . 2. Key Properties Periodicity: $ Z_x(t + T, f) = e^{j 2 \pi f T} Z_x(t, f) $ Time-Frequency Analysis: Converts time-domain signal into a representation showing both time and frequency structure . Often used in Gabor frames, multicarrier communications, and OFDM . Invertible: There’s an inverse Zak Transform to recover x(t) . 4. Discrete Zak ...

MATLAB Code for OTFS (Orthogonal Time Frequency Space) %% Clear workspace clc; clear; close all ; %% Step 1: OTFS Parameters N_delay = 4; % Number of delay bins (rows) N_doppler = 4; % Number of Doppler bins (columns) N_sym = N_delay * N_doppler; modOrder = 4; % QPSK SNR_dB = 20; % Noise level %% Step 2: Generate random data symbols data = randi([0 modOrder-1], N_sym, 1); txSymbols = pskmod(data, modOrder, pi/4); disp( 'Transmitted Delay-Doppler symbols:' ); disp(reshape(txSymbols, N_delay, N_doppler)); %% Step 3: Map Delay-Doppler → Time-Frequency (ISFFT) % ISFFT: Inverse Symplectic Finite Fourier Transform % 1. Take IDFT along Doppler (columns) % 2. Take DFT along Delay (rows) ddSymbols = reshape(txSymbols, N_delay, N_doppler); % Step 3a: IDFT along columns (Doppler) tfGrid = ifft(ddSymbols, N_doppler, 2); %IFFT (accross columns) along Doppler → spreads in time (Delay → Time) %FFT (accross rows)along Delay → spreads in frequency (Delay → Frequency) % Step 3b: DFT along ...

DMRS-Based Channel Estimation in 5G NR 1. System Model (OFDM-Based 5G NR) After CP removal and FFT, the received signal at subcarrier k and OFDM symbol n: $$ Y(k,n) = H(k,n)X(k,n) + W(k,n) $$ Where: X(k,n): transmitted symbol H(k,n): channel frequency response W(k,n) ~ CN(0, σ²): AWGN Y(k,n): received symbol 2. Channel Representation Time-Domain Multipath Channel $$ h(\tau,t) = \sum_{l=0}^{L-1} \alpha_l(t)\delta(\tau - \tau_l) $$ Frequency-Domain Channel $$ H(k,n) = \sum_{l=0}^{L-1} \alpha_l(n)e^{-j2\pi k \Delta f \tau_l} $$ 3. Least Squares (LS) Channel Estimation $$ Y_{DMRS}(k,n) = H(k,n)X_{DMRS}(k,n) + W(k,n) $$ $$ \hat{H}_{LS}(k,n) = \frac{Y_{DMRS}(k,n)}{X_{DMRS}(k,n)} $$ Vector Form $$ \mathbf{Y} = \mathbf{X}\mathbf{H} + \mathbf{W} $$ $$ \hat{\mathbf{H}}_{LS} = (\mathbf{X}^H \mathbf{X})^{-1} \mathbf{X}^H \mathbf{Y} $$ LS Estimator Properties $$ E[\hat{H}_{LS}] = H $$ $$ Var(\hat{H}_{LS}) = \frac{\sigma^2}{|X|^2} $$...

Signal Transformation The signal \( x\!\left(\frac{t-1}{2}\right) \) is a transformed version of \( x(t) \) . It differs from \( x(t) \) in two ways: Time scaling (stretching) Time shifting (delay) 1. Compare the Arguments Original signal: \[ x(t) \] Transformed signal: \[ x\!\left(\frac{t-1}{2}\right) \] The key expression is: \[ \frac{t-1}{2} \] 2. Rewrite the Expression \[ \frac{t-1}{2} = \frac{1}{2}(t-1) \] Step A: Time Scaling (Stretching) The factor 1/2 inside the argument causes time expansion. If \( |a| > 1 \) then the signal compresses If \( 0 < |a| < 1 \) then the signal stretches Since \( a = 1/2 \), the signal is: Stretched by factor 2 Step B: Time Shift (Delay) To find the shift, set: \[ \frac{t-1}{2} = 0 \] Then: \[ t = 1 \] The signal is delayed...

Addition of u(t-1) and u(t-3) To add u(t-1) and u(t-3) , first understand what each term represents. The unit step function is defined as: u(t-a) = 0, t < a 1, t ≥ a Step 1: Understand Each Function 1. u(t-1) u(t-1) = 0, t < 1 1, t ≥ 1 2. u(t-3) u(t-3) = 0, t < 3 1, t ≥ 3 Step 2: Add Them y(t) = u(t-1) + u(t-3) Case 1: t < 1 Both are zero. y(t) = 0 + 0 = 0 Case 2: 1 ≤ t < 3 u(t-1) = 1 and u(t-3) = 0 y(t) = 1 + 0 = 1 Case 3: t ≥ 3 Both are 1. y(t...

Difference Between Impulse Response and Excitation Function The terms impulse response and excitation function are related but refer to different concepts in system analysis (signals and systems, control systems, vibration analysis). 1. Excitation Function (Input) An excitation function is the input signal or force applied to a system to stimulate or excite it. It can be any signal : step, impulse, sinusoid, random noise, etc. It is something you apply to the system . Also called the input , forcing function , or driving function . Examples: A hammer strike on a structure A voltage applied to a circuit A sinusoidal force applied to a spring-mass system y(t) = System { x(t) } Here, x(t) ...

When is a System Stable? A system is stable if its output remains bounded and well-behaved over time , especially when subjected to bounded inputs. 1. BIBO Stability (Bounded-Input Bounded-Output Stability) A system is BIBO stable if: Every bounded input produces a bounded output. If |x(t)| < Mx < ∞ then |y(t)| < My < ∞ For some finite constant My. For LTI Systems: A continuous-time LTI system is stable if and only if : ∫ |h(t)| dt < ∞ That is, the impulse response must be absolutely integrable . For discrete-time systems: Σ |h[n]| < ∞ 2. Stability in Terms of Poles (Control Systems) Continuous-Time System: ...

Matched Filter: Theory and MATLAB Example A matched filter is a linear filter designed to maximize the signal-to-noise ratio (SNR) for detecting a known signal in the presence of additive noise. It is widely used in communications, radar, and sonar . 1. Definition If the known signal is s(t) with duration T , the impulse response of the matched filter is: h(t) = s*(T - t) In discrete time, for a signal s[n] of length N: h[n] = s*[N-1-n] 2. Example: Rectangular Pulse Consider a simple rectangular pulse: s(t) = { 1, 0 ≤ t ≤ 2 0, elsewhere } Matched filter: h(t) = s(2 - t) s(t): ┌───────┐ | | ───────────┘ └──────── 0 2 h(t): ┌───────┐ | | ───────────-------- | 0 2 3. MATLAB Simulation Example We can implement a simple matched filter in MATLAB using a rectangular pulse: % MATLAB Code: Matched Filter Example % Parameters T = 2; ...