Matched Filter Simulator for BPSK Random Bits    ↓ BPSK Mapping (±1)    ↓ Upsampling    ↓ RRC 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...

Difference Between Eigenvalue, Eigenvector and Eigenfunction 1. Eigenvalue and Eigenvector (Linear Algebra) Definition For a square matrix \( A \): \[ A \mathbf{v} = \lambda \mathbf{v} \] \( \mathbf{v} \neq 0 \) → Eigenvector \( \lambda \) → Eigenvalue \( A \) → Linear transformation The matrix scales the vector without changing its direction. Finding Eigenvalues \[ (A - \lambda I)\mathbf{v} = 0 \] For non-trivial solution: \[ \det(A - \lambda I) = 0 \] This is the characteristic equation . Numerical Example ...

Why a Sine Wave is an Eigenfunction of an LTI System 1. What is an Eigenfunction? A function \( x(t) \) is called an eigenfunction of a system if: \[ \text{System}\{x(t)\} = \lambda x(t) \] Where: The output has the same shape as the input It is only scaled (and possibly phase shifted) \( \lambda \) is the eigenvalue 2. LTI System Description An LTI (Linear Time-Invariant) system is completely described by its impulse response \( h(t) \). The output is given by convolution: \[ y(t) = x(t) * h(t) \] \[ y(t) = \int_{-\infty}^{\...

1. Fourier Series of a 50% Duty Cycle Square Wave For a square wave with amplitude A : \[ x(t) = \frac{4A}{\pi}\left[ \sin(\omega t) + \frac{1}{3}\sin(3\omega t) + \frac{1}{5}\sin(5\omega t) + \cdots \right] \] Observations: Only odd harmonics exist (1st, 3rd, 5th, 7th, ...) Amplitude of nth harmonic: \[ X_n = \frac{4A}{n\pi} \quad (n = 1,3,5,7...) \] \[ X_n \propto \frac{1}{n} \] 2. Power of Each Harmonic Power depends on the square of RMS value. \[ X_{n(rms)} = \frac{X_n}{\sqrt{2}} \] \[ ...

Bit Flipping and Equilibrium Theory In digital systems, we often want to bring a binary bit array to a specific equilibrium (e.g., all 1s or all 0s) using minimum operations . If we can flip exactly k bits at a time: Count the number of wrong bits: w . The minimum number of operations required = ceil(w / k) . If w % k ≠ 0 , the last operation may include already-correct bits. Flipping bits in a sliding window approach may require more steps, but choosing any k wrong bits at each step ensures the fewest operations. Example 1 Initial bits: [0, 1, 1, 0] Target equilibrium: all 1s → [1, 1, 1, 1] k = 2 (flip 2 bits at a time) Step 1: Count wrong bits Wrong bits = positions 0 and 3 → total w = 2 Minimum operations = ceil(w / k) = 1  Step 2: Flip 2 wrong bits Operation Bits Before Bits Flipped Bits After 1 [0, 1, 1, 0] [0, 3] [1, 1, 1, 1]  Example 2: Sliding Window Bits: [0, 1, 0, 0, 1] Target: all 1s → [1, 1...

How a Load Balancer Decides Which Server Handles Traffic A load balancer decides which server should handle a request based on a traffic distribution algorithm. Its goal is to distribute traffic efficiently, prevent overload, and improve availability. 1. Basic Process A client sends a request to a website/app. The request first reaches the load balancer (not directly the servers). The load balancer selects one backend server using a predefined method. The request is forwarded to that server. The server responds back through the load balancer to the client. 2. Common Load Balancing Algorithms Round Robin Requests are distributed sequentially. Request 1 → Server A Request 2 → Server B Request 3 → Server C Request 4 → Server A (repeat) Pros: Simple Cons: Doesn’t consider serve...