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FFT Butterfly Method Explained (with Simulations)

4-Point FFT Using Butterfly Method Given: x[n] = {0, 1, 2, 3} Step 1: Split into Even & Odd Even indices: x e = {x[0], x[2]} = {0, 2} Odd indices: x o = {x[1], x[3]} = {1, 3} Step 2: 2-point DFT For any {a, b}: DFT = {a + b, a - b} Even Part (E): {0+2, 0-2} = {2, -2} Odd Part (O): {1+3, 1-3} = {4, -2} Step 3: Combine Using Butterfly X[k] = E[k] + W 4 k O[k] X[k + 2] = E[k] - W 4 k O[k] Twiddle Factors (N=4): W 4 0 = 1, W 4 1 = -j Final Calculations: X[0] = E[0] + W 4 0 O[0] = 2 + (1)(4) = 6 X[2] = E[0] - W 4 0 O[0] = 2 - (1)(4) = -2 X[1] = E[1] + W 4 1 O[1] = -2 + (-j)(-2) = -2 + 2j X[3] = E[1] - W 4 1 O[1] = -2 - (-j)(-2) = -2 - 2j Final Answer: X[k] = {6, -2 + 2j, -2, -2 - 2j} 8-Point FFT Using Butterfly Method Given: x[n] = {0,1,2,3,4,5,6,7} Step 1: Split into Bit-Reversed Order To perform DIT-FFT, split the 8 points into pairs of two: Group A: {x[0], x[4]} = {0, 4}...
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Design of CMOS Flip-Flops (SR, D, JK)

Design of CMOS Flip-Flops (SR, D, JK) A flip-flop or latch is a circuit with two stable states, used to store state information. It is the basic storage element in sequential logic and a fundamental building block in digital electronics systems, including computers and communication devices. Flip-flops and latches act as data storage elements for states, pulse counting, and synchronization of variably-timed input signals to a reference clock. Flip-flops can be transparent/opaque (latches) or clocked (synchronous, edge-triggered). Latches are level-sensitive, while flip-flops are edge-sensitive. In sequential logic, the output depends on current inputs and previous states. Fig.1 shows a sequential circuit combining a combinational block and a memory element. ...
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Pulse Amplitude Modulation and Demodulation

📘 Overview & Theory of Pulse Amplitude Moduation (PAM) 🧮 Pulse Amplitude Demoduation 🧮 MATLAB Code for PAM 📚 Further Reading 📂 Other Topics on Pulse Amplitude Modulation ... 🧮 Simulation results for comparison of PAM, PWM, PPM, DM, and PCM 🧮 Other Pulse Modulation Techniques (e.g., PWM, PPM, DM, and PCM) 🧮 MATLAB Code for Pulse Amplitude Modulation and Demodulation of an Analog Signal (2) 🧮 MATLAB Code for Pulse Amplitude Modulation and Demodulation of Digital data  Pulse Amplitude Modulation (PAM) Sampling allow us to represent real world continuous signal, such as audio or video, in a format suitable for digital processing and storage. This sampled discrete-time signal is inherently digital. A digital signal is a discrete-time signal that is further quantized in amplitude. Pulse Amplitude modulation (PAM) is the modulation technique in which amplitude of carrier pulses is...
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MATLAB Code for BER performance of QPSK with BPSK, 4-QAM, 16-QAM, 64-QAM, 256-QAM, etc

📘 Overview 🧮 MATLAB Codes 🧮 Online Simulator for Calculating BER of M-ary PSK and QAM 🧮 QPSK vs BPSK and QAM: A Comparison of Modulation Schemes in Wireless Communication 🧮 Are QPSK and 4-PSK same? 📚 Further Reading   QPSK offers double the data rate of BPSK while maintaining a similar bit error rate at low SNR when Gray coding is used. It shares spectral efficiency with 4-QAM and can outperform 4-QAM or 16-QAM in very noisy channels. QPSK is widely used in practical wireless systems, often alongside QAM in adaptive modulation schemes [Read more...] What is the Gray Code? Gray Code: Gray code is a binary numeral system where two successive values differ in only one bit. This property is called the single-bit difference or unit distance code. It is also known as reflected binary code. Let's convert binary 111 to Gray code: Binary bits: B = 1 1 1 Apply the rule: G[0] = B[0] = 1...
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MATLAB Code for QPSK Modulation and Demodulation

📘 Overview 🧮 MATLAB Codes 🧮 Theory 🧮 BER performance of QPSK with BPSK, 4-QAM, 16-QAM, 64-QAM, 256-QAM, etc 📚 Further Reading QPSK Passband Signal Generation Spectral Efficiency in QPSK   Quadrature Phase Shift Keying (QPSK) is a digital modulation scheme that conveys two bits per symbol by changing the phase of the carrier signal. Each pair of bits is mapped to one of four possible phase shifts: 0°, 90°, 180°, or 270° 00  ===> 0 degree phase shift of carrier signal 01  ===> 90 degree 11  ===> 180 degree 10  ===> 270 degree   MATLAB Script clc; clear all; close all; clc; M = 4; data = randi([0 (M-1)], 1000, 1); Phase = 0; modData=pskmod(data,M,Phase); figure(1); scatterplot(modData); channelAWGN = 15; rxData2 = awgn(modData, channelAWGN); figure(2); scatterplot(rxData2); demodData = pskdemod(rxData2,M,Phase);   Result data 1 0 2 2 0 2 1 . . . modData -1.0...
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BER vs SNR for M-ary QAM, M-ary PSK, QPSK, BPSK, ...(MATLAB Code + Simulator)

Bit Error Rate (BER) & SNR Guide Analyze communication system performance with our interactive simulators and MATLAB tools. 📘 Theory 🧮 Simulators 💻 MATLAB Code 📚 Resources BER Definition SNR Formula BER Calculator MATLAB Comparison 📂 Explore M-ary QAM, PSK, and QPSK Topics ▼ 🧮 Constellation Simulator: M-ary QAM 🧮 Constellation Simulator: M-ary PSK 🧮 BER calculation for ASK, FSK, and PSK 🧮 Approaches to BER vs SNR Calculation What is Bit Error Rate (BER)? The BER indicates how many corrupted bits are received compared to the total number of bits sent. It is the primary figur...
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Frequency Bands : EHF, SHF, UHF, VHF, HF, MF, LF, VLF and Their Uses

Frequency Bands >> EHF, SHF, UHF, VHF, HF, MF, LF... Frequency Bands and Their Uses 1. Extremely High Frequency (EHF) 30 - 300 GHz Uses 5G Networks 5G millimeter wave band 6G and beyond (Experimental) RADAR 2. Super High Frequency (SHF) 3 - 30 GHz Uses Ultra-wideband (UWB) Airborne RADAR Satellite Communication Microwave Link Communication or SATCOM 3. Ultra High Frequency (UHF) 300 - 3000 MHz Uses Satellite Communication Television Surveillance Navigation aids Also, read important wireless communication terms 4....
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FM Bandwidth and FM Band Explained

FM radio uses the frequency band from 88 MHz to 108 MHz , which is a 20 MHz-wide spectrum . This is the range of carrier frequencies available to stations. 108 MHz − 88 MHz = 20 MHz However, a single FM station occupies only about 200 kHz . This is the bandwidth of the modulated FM signal. 1. Why One FM Station Needs ~200 kHz FM uses frequency modulation . The bandwidth depends on how far the carrier swings. Carson's Rule gives the approximate FM bandwidth: B = 2 ( Δf + f m ) ...
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