In signal processing, most transform basis functions look like \( e^{(\text{something})} \) — such as \( e^{j\omega t} \), \( e^{-st} \), or \( e^{-j2\pi kt/N} \). This is not a coincidence. Exponentials have deep mathematical properties that make them ideal for representing and analyzing signals. 1. Exponentials Are Eigenfunctions of LTI Systems If you feed an LTI system a complex exponential: x(t) = e^{st} the output has the same shape: y(t) = H(s) e^{st} This makes exponentials extremely convenient because they diagonalize LTI systems and turn convolution into simple multiplication. 2. Sines and Cosines Are Special Cases of Complex Exponentials Euler’s identity: e^{j\omega t} = \cos(\omega t) + j\sin(\omega t) ...

Fourier Series Calculator Choose a Signal Rectangular Wave Square Wave (50% Duty) Sawtooth Wave Triangular Wave Sine Wave (Example) Parameters Amplitude A: Period T: Pulse Width (for rectangular only): Number of Harmonics (N): Compute Fourier Series MATLAB Code syms t n % Parameters T = 4; % period w0 = 2*pi/T; % fundamental frequency n = 1:7; % harmonic index (symbolic sum up to 7) % Define the sine wave f = sin(w0 * t); % Fourier coefficients a0 = (1/T) * int(f, t, 0, T) an = (2/T) * int(f * cos(n*w0*t), t, 0, T) bn = (2/T) * int(f * sin(n*w0*t), t, 0, T) Output a0 = 0 an = [0, 0, 0, 0, 0, 0, 0] bn = [1, 0, 0, 0, 0, 0, 0] Further Reading

  Passband versions of Amplitude Shift Keying (ASK), Frequency Shift Keying (FSK), and Phase Shift Keying (PSK) Passband modulation techniques involve shifting the baseband signal to higher frequencies before transmission. Here are the passband versions of Amplitude Shift Keying (ASK), Frequency Shift Keying (FSK), and Phase Shift Keying (PSK):     Passband Amplitude Shift Keying (PASK):         In PASK, the baseband signal is modulated onto a carrier signal by varying its amplitude.         The modulated signal is then shifted to a higher frequency band (passband) for transmission.         PASK is achieved through the process of mixing or multiplying the baseband signal with a carrier signal.         The demodulation process involves extracting the amplitude variations of the passband signal to recover the original baseband sign...

Convolutions in Neural Networks Convolutions let neural networks efficiently detect local patterns everywhere in an image without needing a huge, complex weight matrix. Instead of manually creating a complicated matrix, use a convolution, which is a local, translation-invariant linear operation. A kernel (small weight matrix, like 3×3) is multiplied with each neighborhood of the image, sliding across the image. The same kernel weights are reused across the entire image, so the network can detect the same pattern anywhere. Kernel weights are initialized randomly and learned via backpropagation. Each weight in the kernel contributes to outputs at multiple locations, so gradients come from the whole image. 1. What a pattern is A pattern is just a small arrangement of pixels. For example, in a black-and-white image: 1 1 0 1 0 0 This could be part of an edge or corner in an image. The ne...

    UGC-NET Electronic Science Question Paper With Answer Key Download Pdf [2023] Download Question Paper               See Answers   2025 | 2024 | 2023 | 2022 | 2021 | 2020 UGC-NET Electronic Science  2023 Answers with Explanations Q.118 Let's analyze and solve the question. List-I (Sequence x[n]) (A) \( e^{i \omega_0 n} x[n] \) (B) \( n x[n] \) (C) \( x[n] - x[n-1] \) (D) \( \delta[n - n_0] \) List-II (Fourier Transform X(Ω)) (I) \( (1 - e^{-i \Omega}) \times (\Omega) \) (II) \( X(\Omega - \Omega_0) \) (III) \( e^{-i \Omega n_0} \) (IV) \( \frac{dX(\Omega)}{d\Omega} \) Matching the sequences with their Fourier Transforms: For \( e^{i \omega_0 n} x[n] \): Multiplying the sequence \( x[n] \) by a complex exponential will shift its Fourier transform b...

Time Division Multiple Access (TDMA) Overview Time Division Multiple Access (TDMA) is a channel access method used in communication systems to allow multiple users to share the same frequency channel. Each user is allocated a specific time slot during which they can transmit or receive data. TDMA is used in cellular networks (e.g., GSM), satellite communications, and other wireless systems. Key Concepts of TDMA Time Slots: TDMA divides the available bandwidth into discrete time slots, each allocated to a user. Each user transmits during their allocated time slot. Frame Structure: The total available time is organized into frames, where each frame consists of several time slots. A frame has slots for each user. Channel Sharing: Multiple users share the same frequency by transmitting in different time slots. Mathematical Representation of TDMA ...

Poles and Zeros in Signal Processing 1. Poles Poles are values of z (or s in analog) that make the filter’s denominator zero , i.e., locations where the filter’s transfer function theoretically goes to infinity. Digital filter example: H(z) = B(z) / A(z)     // Poles are roots of A(z) = 0 Effect of poles: Correspond to resonant frequencies Close to the unit circle → narrowband, high amplification Far from the unit circle → damped, broad frequency response Poles essentially generate/boost frequency components Intuition: Poles create peaks in the frequency spectrum (resonances). 2. Zeros Zeros are values of z (or s ) that make the numerator ze...

Why [1, -1] is a High-Pass Filter 1. What is a High-Pass Filter? A high-pass filter (HPF) is a filter that allows high-frequency components to pass while attenuating (reducing) low-frequency components . Low frequency → smooth/slow changes High frequency → rapid/fast changes 2. The Filter [1, -1] Consider a 1D discrete signal \( x[n] \). If we apply the filter [1, -1], the output \( y[n] \) is: \[ y[n] = 1 \cdot x[n] + (-1) \cdot x[n-1] = x[n] - x[n-1] \] 3. What does it do? It computes the difference between consecutive samples . If the signal is constant (low frequency, smooth), then \( x[n] - x[n-1] \approx 0 \) → output is near zero. If the signal changes quickly (high frequency), then \( x[n] - x[n-1] \) is large → output is large. This is exactly what a high-pass filter does. ...