Suppose, you have two arrays A = [1,2,3,4], and B = [10,11,12]. Then the convolution result, C will be [10,31,64,97,80,80]. The length of C is length of A + length of B -1 . Convolution is a very useful concept in signal processing. We've often taken consideration that transmitted signal is convolved with channel impulse response (CIR) in the time domain. And in the frequency domain they are in multiplication form. Lets discuss how mathematically find the convolution of two arrays or signals. For the above example, the formula for convolution is: \[ C[n] = (A * B)[n] = \sum_{m= 0}^{m} A[m] B[n - m] \]  Where, m is the length of A      and, n varies from 0 to m + p -1   where, p is the length of B   Convolution Formula The convolution of two continuous-time signals \( x(t) \) and \( h(t) \) is given by: \[ y(t) = (x * h)(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) \, d\tau \] For discrete-time signals

  Baseband modulation refers to the process of transmitting digital signals over a communication channel without changing the frequency of the signal to a higher-frequency carrier. The signal is transmitted directly without frequency translation, making it suitable for short distances, such as within a local area network (LAN). Baseband Amplitude Shift Keying (ASK) For Binary Amplitude Shift Keying (BASK), binary bit '0' can be represented as lower level voltage and bit '1' as higher level voltage. For example, you can map binary bit '0' to 0 and bit '1' to 5 volts. Then, you can transmit the signal to the wire to the destination.  Baseband Frequency Shift Keying (FSK) For Binary Frequency Shift Keying (BFSK), you can map binary bit '0' to 'j' and bit '1' to '1'. Here, signals are in phase. But it is not necessary for BFSK. For baseband representation, we can map the bits like these for various purposes, such as calculatin

  Signal normalization is common in signal processing, especially when a signal emerges from a filtering process. For example, you can normalize a highly attenuated filtered signal to an amplitude range of -1 to 1. MATLAB Script % Parameters for the sine wave fs = 1000; % Sampling frequency t = 0:1/fs:1; % Time vector f = 15; % Frequency of the sine wave % Example signal: Noisy sine wave filtered_signal = 0.03*sin(2 * pi * f * t) + 0.05*sin(2 * pi * f * t); % Step 2: Normalize the filterd signal to the range [-1, +1] normalized_signal = (filtered_signal - min(filtered_signal)) / (max(filtered_signal) - min(filtered_signal)); normalized_signal = normalized_signal * 2 - 1; % Scale to [-1, +1] % Original Signal figure(); plot(t, filtered_signal, 'b', 'LineWidth', 1.5); title('Filtered Signal'); xlabel('Time (s)'); ylabel('Amplitude'); ylim([-0.1 0.1]); grid on; % Normalized Signal figure(); plot(t, normalized_signal, 'g', 'LineWidth'

Explore the concept of Quantization Signal-to-Noise Ratio (SNR), a critical parameter in Pulse Code Modulation (PCM) that determines the fidelity of quantized signals in digital communication systems. Core Concepts of Quantization SNR Definition of Quantization SNR Quantization SNR measures the ratio of the power of the quantized signal to the power of the quantization noise introduced during the quantization process. Psnr = Ps / Pq, Or, Psnr = Ps / (Δ² / 12)   Where Psnr is the quantization SNR, Ps is the average power of the signal, Pq is the quantization noise power, and Δ is the quantization step size. Importance in PCM In PCM systems, high quantization SNR ensures better signal reconstruction at the receiver, leading to improved quality and performance. Factors Affecting Quantization SNR

  MATLAB Script clc; clear; close all; % SNR range in dB SNRdB = 0:2:20; % Convert SNR from dB to linear scale EbN0Lin = 10.^(SNRdB / 10); % Theoretical BER for 2x2 Alamouti with BPSK modulation pAlamouti = 1/2 - 1/2*(1+2./EbN0Lin).^(-1/2); theoryBerAlamouti_nTx2_nRx1 = pAlamouti.^2.*(1+2*(1-pAlamouti)); % Plot the theoretical BER vs SNR figure; semilogy(SNRdB, theoryBerAlamouti_nTx2_nRx1, '*-r', 'LineWidth', 2); grid on; xlabel('SNR (dB)'); ylabel('Bit Error Rate (BER)'); title('Theoretical BER vs SNR for 2x2 Alamouti Scheme with BPSK'); legend('Theoretical BER'); Output  Copy the MATLAB Code from Here % The code is written by SalimWireless.Com clc; clear; close all; % Parameters bit_stream = [1, 1, 0, 0, 1, 0, 1, 1, 1, 0]; % Original bit stream N = length(bit_stream); % Number of samples filter_order = 10; % Order of the adaptive filter lambda = 0.99; % Forgetting factor for RLS algorithm delta = 1; % Initial value for the inverse c

Transmitted and Received Power in Communication Systems Key Concepts Transmitted Power ( P s ): The average power of the transmitted signal is defined as the power that is radiated from the transmitter. For a BPSK signal, where the transmitted symbols are either +1 or -1 , the average transmitted power is: P s = 1 Since the magnitude of both +1 and -1 is 1 , the average power of the transmitted BPSK signal is 1 . Channel Coefficient ( h ): The channel coefficient h characterizes how the transmitted signal is affected as it propagates through the channel. It can be represented as a complex number: h = a + jb The magnitude squared of the channel coefficient |h|² gives us the channel gain, which describes how much the signal power is amplified or attenuated as it passes through the channel: |h|

AWGN Noise: Mean and Variance in Practical Systems In practical communication systems, Additive White Gaussian Noise (AWGN) is modeled with a zero mean and variance based on the signal power and signal-to-noise ratio (SNR). AWGN Mean In most practical systems, the mean of the AWGN noise is set to zero. This is because AWGN is symmetric around zero, making it equally likely to increase or decrease the signal. Why Zero Mean? A zero mean ensures that the noise doesn’t introduce a consistent bias to the signal. Mean of AWGN = 0 AWGN Variance The variance of AWGN is determined based on the signal power and the desired SNR. 1. SNR Definition: SNR = P signal / P noise Where: P signal is the average power of the signal. P noise is the power (variance) of the noise. 2. Noise Variance: P noise = σ 2 3. Convert SNR from dB to

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