Why Complex Exponentials Are Eigenfunctions of Every LTI System Eigenfunction of an LTI System For any LTI system , complex exponentials are eigenfunctions. If the input is: $$ x(t) = e^{j\omega_0 t} $$ then the output is: $$ y(t) = H(j\omega_0)\, e^{j\omega_0 t} $$ where \(H(j\omega_0)\) is the system’s frequency response. Why is it called an eigenfunction? Because it satisfies the eigenvalue equation: $$ T\{x(t)\} = \lambda x(t) $$ Eigenfunction → \( e^{j\omega_0 t} \) Eigenvalue → \( H(j\omega_0) \) So the eigenvalue is not \( e^{j\omega_0 t} \). The eigenvalue is the scalar \( H(j\omega_0) \). What about...

RLC Network Question and Solution Question A network consisting of a finite number of linear resistor (R), inductor (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form: Σ (k = 1 to 3) aₖ cos(kω₀ t), where aₖ ≠ 0, ω₀ ≠ 0. The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network? Σ (k = 1 to 3) bₖ cos(kω₀ t + φₖ), where bₖ ≠ aₖ for all k ...

  In OFDM (Orthogonal Frequency Division Multiplexing) , we transmit multiple orthogonal subcarriers simultaneously. Since the subcarriers are orthogonal , they do not interfere with each other, which is one of the main advantages of OFDM. Practically, OFDM converts a wideband signal into multiple narrowband orthogonal subcarriers. For typical wireless communication, if the signal bandwidth (or symbol duration) exceeds the coherence bandwidth of the channel, the signal experiences frequency-selective fading . Fading distorts the signal, making it difficult to recover the original information. By using OFDM, we transmit the same wideband signal across multiple orthogonal narrowband subcarriers, reducing the effect of fading. For example, if we want to transmit a signal of bandwidth 1024 kHz , we can divide it into N = 8 subcarriers . Each subcarrier is then spaced by: Δf = Total Bandwidth N = 1024 8 kHz...

LU Decomposition using Doolittle Factorization We can write an m X n matrix A as a product of two matrices, L and U . And A = L*U L = $\ \begin{bmatrix} 1 & 0 & 0 & 0 \\ l21 & 1 & 0 & 0 \\ l31 & l32 & 1 & 0 \\ l41 & l42 & l43 & 1 \end{bmatrix}$ ; U = $\begin{bmatrix} u11 & u12 & u13 & u14 \\ 0 & u22 & u23 & u24 \\ 0 & 0 & u33 & u34 \\ 0 & 0 & 0 & u44 \end{bmatrix}$ L= lower triangular matrix; U= upper triangular matrix Procedure- Choose a matrix ( m X n) (e.g., 3X 3, 3 X 4, 4 X 4, etc.,) Initialize the L and U matrices. For L matrix, take a matrix with all diagonal elements assigned to 1, and the matrix elements above the diagonal are zeroes. L matrix size will be ( m X m ). The values of matrix elements below the main diagonal can be assigned to l 21 , l 31 , etc., and so on. $$\ \begin{bmatrix} 1 & 0 & 0 & 0 \\ l21 & 1 & 0 & 0 \\ l31 & l32 ...

LU Decomposition using Crout’s Method Procedure- Choose a matrix ( m X n) (e.g., 3X 3, 3 X 4, 4 X 4, etc.,) The Crout’s matrix decomposition algorithm differs slightly from the Doolittle method. Doolittle's method returns a unit lower triangular matrix and an upper triangular matrix, while the Crout’s method returns a lower triangular matrix and a unit upper triangular matrix. Initialize the L and U matrices. L matrix size will be ( m X m ). The values of matrix elements below the main diagonal can be assigned to l 21 , l 31 , etc., and so on. And the matrix elements above the diagonal are zeroes. $$\ \begin{bmatrix} l11 & 0 & 0 & 0 \\ l21 & l22 & 0 & 0 \\ l31 & l32 & l33 & 0 \\ l41 & l42 & l43 & l44 \end{bmatrix} $$ l 21 , l 31 , etc. are unknown For matrix U , take a matrix with all diagonal elements assigned to 1, and the matrix elements below the diagonal are zeroes. Size of matrix U will be as same as matrix A (...

Discriminator Receiver Where is a Discriminator Receiver Used? It is used in receivers for: Frequency Modulation (FM) signals FM broadcast radio systems Two-way radios (walkie-talkies) TV sound transmission (analog TV systems) What Does a Discriminator Do? In FM: Information is carried by frequency changes of the carrier. The amplitude remains constant . A discriminator: Converts frequency variations into voltage variations . ...

Frequency Shift Keying (FSK) FSK is a digital modulation technique where each bit is represented by a different carrier frequency: 0 → f 0 ,   1 → f 1 The transmitted FSK signal can be represented as: s(t) = A cos(2π f i t),   i ∈ {0,1} When transmitted over a noisy channel, Additive White Gaussian Noise (AWGN) spreads the received signal. This simulation shows the effect of noise on the FSK constellation by plotting each symbol based on its frequency and amplitude. Number of Random Bits: Signal-to-Noise Ratio (SNR) in dB:

Binary Phase Shift Keying (BPSK) Simulation BPSK is a type of digital modulation where each bit is represented by one of two possible phases of a carrier signal: 0 → -1,   1 → +1 The transmitted BPSK signal can be written as: s(t) = √(2E_b) * cos(2πf_c t + πb),   b ∈ {0,1} When transmitted over a noisy channel, Additive White Gaussian Noise (AWGN) affects the signal. At the receiver, the demodulated signal is compared with a threshold (0) to detect the bit. This simulation demonstrates the effect of noise on BPSK constellation points and allows you to calculate the Bit Error Rate (BER). Number of Bits: SNR (dB):