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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

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MATLAB Code for BER performance of QPSK with BPSK, 4-QAM, 16-QAM, 64-QAM, 256-QAM, etc

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MATLAB Code for QPSK Modulation and Demodulation

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BER vs SNR for M-ary QAM, M-ary PSK, QPSK, BPSK, ...(MATLAB Code + Simulator)

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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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