User Input +---------------------------+ | # Subcarriers (Nsc)       | | # OFDM Symbols (Nsym)     | | DMRS length (Ldmrs)       | | # Channel taps            | | Noise std deviation       | +---------------------------+               |               v Generate DMRS Pilot Sequence +---------------------------+ | QPSK symbols of length Ldmrs | e.g., [1+j, -1+j, ...] +---------------------------+               |               v Frequency-Domain Mapping +---------------------------+ | Zero-pad DMRS to all Nsc   <-- Pilot insertion | DMRS on first Ldmrs, rest 0 | X[k] = [DMRS, 0, 0, ..., 0] +---------------------------+               |               v OFDM Modu...

What is BLER (Block Error Rate)? Definition BLER (Block Error Rate) is the percentage of transmitted data blocks that are received incorrectly and fail error detection (CRC check) after decoding. It is a key performance metric in 4G and 5G wireless systems. Formula BLER = (Number of incorrectly received blocks) / (Total transmitted blocks) Example: If 100 transport blocks are transmitted and 10 fail CRC: BLER = 10 / 100 = 10% What is a Block? Data is transmitted in Transport Blocks (TB) Each TB contains multiple code blocks (LDPC coded) If CRC fails, the entire TB is considered erroneous Why is BLER Important? Determines throughput A...

Cooley–Tukey FFT vs Conventional FFT Algorithm Understanding the difference between the Cooley–Tukey Fast Fourier Transform and the conventional FFT/DFT computation. 1. Conventional FFT Algorithm The conventional FFT algorithm usually refers to the direct computation of the Discrete Fourier Transform (DFT) . The DFT of a sequence of length \(N\) is: \[ X[k] = \sum_{n=0}^{N-1} x[n] e^{-j 2\pi nk / N}, \quad k = 0,1,\dots,N-1 \] Direct DFT computation requires O(N²) operations (multiplications and additions) and can be slow for large sequences. 2. Cooley–Tukey FFT Algorithm The Cooley–Tukey algorithm is a widely used FFT method based on divide-and-conquer . It splits a DFT of size \(N\) into smaller DFTs, which reduces computation to O(N log N) . For a sequence of length \(N\) (ideally a power of 2), the algorithm splits into even and odd indices: \[ X[k] = \sum_{n...

Duality Property of the Fourier Transform The duality property (or dual property) of the Fourier transform states: If a function and its Fourier transform form a pair, then swapping them (with a sign change) also forms a valid Fourier transform pair. 1. Statement (Physics Convention) Using the convention: \[ \mathcal{F}\{f(x)\}(k) = \int_{-\infty}^{\infty} f(x)\, e^{-ikx}\, dx \] If \[ f(x) \longleftrightarrow F(k) \] then duality says: \[ F(x) \longleftrightarrow 2\pi\, f(-k) \] The roles of space/time and frequency are exchanged (with a sign flip and a factor \(2\pi\)). 2. Statement (Symmetric / Engineering Convention) Using the symmetric definition: \[ \mathcal{F}\{f(x)\}(k) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i k x}\, dx \] Then duality becomes: \[ f(x) \longleftrightarrow F(k) \quad \Rightarrow \quad F(x) \longleftrightarrow f(-k) \] No extra constant appears. 3. Intuition The Fourier transform is nearly symmetri...

Fourier Transform of a Gaussian The Fourier transform of a Gaussian is another Gaussian . This is one of the most elegant results in analysis and physics. 1. Standard Example Let \[ f(x) = e^{-a x^2}, \quad a > 0 \] Using the physics convention: \[ \mathcal{F}\{f(x)\}(k) = \int_{-\infty}^{\infty} e^{-a x^2} e^{-ikx}\, dx \] The result is: \[ \mathcal{F}\{e^{-a x^2}\}(k) = \sqrt{\frac{\pi}{a}} \; e^{-\frac{k^2}{4a}} \] 2. Key Insight A Gaussian in position space transforms into a Gaussian in frequency space: \[ e^{-a x^2} \quad \longleftrightarrow \quad e^{-\frac{k^2}{4a}} \] Narrow in \(x\) (large \(a\)) → Wide in \(k\) Wide in \(x\) (small \(a\)) → Narrow in \(k\) This is a direct manifestation of the uncertainty principle. 3. Special Case (Unit Gaussian) \[ f(x) = e^{-x^2} \] \[ \mathcal{F}\{e^{-x^2}\}(k) = \sqrt{\pi}\, e^{-k^2/4} \] 4. Normalized Gaussian (Self-Fourier Form) If we choose the symmetric normalizati...

Matched Filter Simulator for BPSK Random Bits    ↓ BPSK Mapping (±1)    ↓ Upsampling    ↓ RRC ((Root Raised Cosine) Filter Pulse Shaping    ↓ Add AWGN Noise    ↓ Matched Filter    ↓ Peak Detection Number of Symbols: Samples per Symbol (Oversampling): Noise Std Dev: Simulate

Fourier Transform: Phase and Magnitude Sign Flip 1. Fourier Transform Basics For a signal x(t) , its Fourier transform X(f) can be written in polar form: X(f) = |X(f)| e^{j φ(f)} Where: |X(f)| is the magnitude spectrum φ(f) = arg(X(f)) is the phase spectrum 2. Changing the Sign of the Magnitude If we flip the sign of the magnitude: X_new(f) = -|X(f)| e^{j φ(f)} This is equivalent to: - |X(f)| e^{j φ(f)} = |X(f)| e^{j (φ(f) + π)} Key Insight: Changing the sign of the magnitude adds π (180°) to the phase. 3. Deep Meaning Sign flips in frequency → phase shift: Each frequency rotates 180° in the complex plane. Signal reconstruction: Inverse Fourier transform of X(f) vs -X(f) gives x(t) vs -x(t) . Intuition: Magnitude shows "how much" of each frequency is present, phase shows "how to align" them. Flipping the sign of magnitude inverts the signal in time domain. 4. Example Suppose X(f) = 2 e^{j...

Applications of Eigenvalues in Signal Processing Eigenvalues are not just theory — they are central to many core signal processing problems. Below I will explain the main practical applications , with proper mathematics and how they are used in real systems. 1. Principal Component Analysis (PCA) – Signal Compression and Denoising Problem Given noisy signal vectors: \[ \mathbf{x}_1, \mathbf{x}_2, ..., \mathbf{x}_N \] Reduce dimensionality Remove noise Keep maximum signal energy Step 1: Form Covariance Matrix \[ R_x = E[\mathbf{x}\mathbf{x}^T] \] This matrix contains signal correlation information. Step 2: Eigenvalue Decomposition \[ R_x \mathbf{v}_i = \lambda_i \mathbf{v}_i \] \( \lambda_i \) = eigenvalues \( \mathbf{v}_i \) = eigenvectors Interpretation Large eigenvalue indicates direction of high signal energy Small eigenvalue indicates mostly noise Practical Use \[ \mathbf{x}_{approx} = \sum_{i=1}^{k} (\mathbf{v}_i^T \mathb...