Skip to main content

Frequency Selective Fading vs Flat Fading in MATLAB


In the MATLAB code below, a comparison between frequency-selective fading and flat fading is shown.
In frequency-selective fading, multipath propagation causes multiple delayed copies of the signal to arrive at the receiver. When the channel delay spread exceeds the symbol duration, these delayed components overlap, resulting in inter-symbol interference (ISI). In flat fading, ISI does not occur because the signal bandwidth is much smaller than the channel’s coherence bandwidth. Therefore, the channel response remains approximately constant across the signal bandwidth, and all symbols experience the same fading.

MATLAB Code for frequency selective fading channel


% OFDM over frequency selective Rayleigh fading channel
clc; clearvars; close all;
% Simulation parameters
nSym = 10^4; % Number of OFDM symbols
EbN0dB = 0:2:20; % Eb/N0 range
MOD_TYPE = 'MPSK'; % 'MPSK' or 'MQAM'
M = 4; % QPSK
N = 64; % Total number of subcarriers
Ncp = 16; % Cyclic prefix length
L = 10; % Number of channel taps
k = log2(M); % Bits per symbol
EsN0dB = 10*log10(k*(N/(N+Ncp))) + EbN0dB; % account CP loss
errors = zeros(1,length(EsN0dB));
% Monte Carlo simulation
for i = 1:length(EsN0dB)
for j = 1:nSym
% -------- Transmitter --------
d = ceil(M*rand(1,N)); % Random symbols
[X , ref] = modulation_mapper(MOD_TYPE,M,d);
x = ifft(X,N); % IDFT
s = add_cyclic_prefix(x,Ncp); % Add CP
% -------- Channel (Freq-selective Rayleigh) --------
h = 1/sqrt(2)*(randn(1,L)+1i*randn(1,L)); % CIR
H = fft(h,N); % Freq response
hs = conv(h,s); % Channel filtering
r = awgn(hs,EsN0dB(i)); % Add noise
% -------- Receiver --------
y = remove_cyclic_prefix(r,Ncp,N); % Remove CP
Y = fft(y,N); % DFT
V = Y ./ H; % Equalization
[~, dcap] = iqOptDetector(V,ref);
errors(i) = errors(i) + sum(d ~= dcap);
end
end
% SER results
simulatedSER = errors/(nSym*N);
theoreticalSER = ser_rayleigh(EbN0dB,MOD_TYPE,M);
% Plot
semilogy(EbN0dB,simulatedSER,'ko'); hold on;
title(['Performance of ',num2str(M),'-',MOD_TYPE,...
' OFDM over Freq Selective Rayleigh channel']);
xlabel('Eb/N0 (dB)');
ylabel('Symbol Error Rate');
function [X, ref] = modulation_mapper(MOD_TYPE, M, d)
% Modulation mapper for OFDM transmitter
% MOD_TYPE - 'MPSK' or 'MQAM'
% M - Modulation order (BPSK=2, QPSK=4, 16-QAM=16, etc.)
% d - Data symbols drawn from {1,2,...,M}
%
% Outputs:
% X - Modulated complex symbols
% ref - Ideal constellation points (for IQ detector)
if strcmpi(MOD_TYPE, 'MPSK')
[X, ref] = mpsk_modulator(M, d); % M-PSK modulation
elseif strcmpi(MOD_TYPE, 'MQAM')
[X, ref] = mqam_modulator(M, d); % M-QAM modulation
else
error('Invalid modulation type. Use ''MPSK'' or ''MQAM''.');
end
end
function [s,ref]=mpsk_modulator(M,d)
%Function to MPSK modulate the vector of data symbols - d
%[s,ref]=mpsk_modulator(M,d) modulates the symbols defined by the
%vector d using MPSK modulation, where M specifies the order of
%M-PSK modulation and the vector d contains symbols whose values
%in the range 1:M. The output s is the modulated output and ref
%represents the reference constellation that can be used in demod
ref_i= 1/sqrt(2)*cos(((1:1:M)-1)/M*2*pi);
ref_q= 1/sqrt(2)*sin(((1:1:M)-1)/M*2*pi);
ref = ref_i+1i*ref_q;
s = ref(d); %M-PSK Mapping
end
function s = add_cyclic_prefix(x,Ncp)
%function to add cyclic prefix to the generated OFDM symbol x that
%is generated at the output of the IDFT block
% x - ofdm symbol without CP (output of IDFT block)
% Ncp-num. of samples at x's end that will copied to its beginning
% s - returns the cyclic prefixed OFDM symbol
s = [x(end-Ncp+1:end) x]; %Cyclic prefixed OFDM symbol
end
function y = remove_cyclic_prefix(r,Ncp,N)
%function to remove cyclic prefix from the received OFDM symbol r
% r - received ofdm symbol with CP
% Ncp - num. of samples at beginning of r that need to be removed
% N - number of samples in a single OFDM symbol
% y - returns the OFDM symbol without cyclic prefix
y=r(Ncp+1:N+Ncp);%cut from index Ncp+1 to N+Ncp
end
function [idealPoints,indices]= iqOptDetector(received,ref)
%Optimum Detector for 2-dim. signals (MQAM,MPSK,MPAM) in IQ Plane
%received - vector of form I+jQ
%ref - reference constellation of form I+jQ
%Note: MPAM/BPSK are one dim. modulations. The same function can be
%applied for these modulations since quadrature is zero (Q=0).
x=[real(received); imag(received)]';%received vec. in cartesian form
y=[real(ref); imag(ref)]';%reference vec. in cartesian form
[idealPoints,indices]= minEuclideanDistance(x,y);
end
function [idealPoints,indices]= minEuclideanDistance(x,y)
%function to compute the pairwise minimum Distance between two
%vectors x and y in p-dimensional signal space and select the
%vectors in y that provides the minimum distances.
% x - a matrix of size mxp
% y - a matrix of size nxp. This acts as a reference against
% which each point in x is compared.
% idealPoints - contain the decoded vector
% indices - indices of the ideal points in reference matrix y
[m,p1] = size(x);[n,p2] = size(y);
if p1~=p2
error('Dimension Mismatch: x and y must have same dimension')
end
X = sum(x.*x,2);
Y = sum(y.*y,2)';
d = X(:,ones(1,n)) + Y(ones(1,m),:) - 2*x*y';%Squared Euclidean Dist.
[~,indices]=min(d,[],2); %Find the minimum value along DIM=2
idealPoints=y(indices,:);
indices=indices.';
end
function [ser] = ser_rayleigh(EbN0dB,MOD_TYPE,M)
%Compute Theoretical Symbol Error rates for MPSK or MQAM modulations
%EbN0dB - list of SNR per bit points
%MOD_TYPE - 'MPSK' or 'MQAM'
%M - Modulation level for the chosen modulation
% - For MPSK M can be any power of 2
% - For MQAM M must be even power of 2 (square QAM only)
gamma_b = 10.^(EbN0dB/10); %SNR per bit in linear scale
gamma_s = log2(M)*gamma_b; %SNR per symbol in linear scale
switch lower(MOD_TYPE)
case {'bpsk'}
ser = 0.5*(1-sqrt(gamma_b/(1+gamma_b)));
case {'mpsk','psk'}
ser = zeros(size(gamma_s));
for i=1:length(gamma_s), %for each SNR point
g = sin(pi/M).^2;
fun = @(x) 1./(1+(g.*gamma_s(i)./(sin(x).^2))); %MGF
ser(i) = (1/pi)*integral(fun,0,pi*(M-1)/M);
end
case {'mqam','qam'}
ser = zeros(size(gamma_s));
for i=1:length(gamma_s) %for each SNR point
g = 1.5/(M-1);
fun = @(x) 1./(1+(g.*gamma_s(i)./(sin(x).^2)));%MGF
ser(i) = 4/pi*(1-1/sqrt(M))*integral(fun,0,pi/2)-4/pi*(1-1/sqrt(M))^2* ...
integral(fun,0,pi/4);
end
case {'mpam','pam'}
ser = zeros(size(gamma_s));
for i=1:length(gamma_s) %for each SNR point
g = 3/(M^2-1);
fun = @(x) 1./(1+(g.*gamma_s(i)./(sin(x).^2)));%MGF
ser(i) = 2*(M-1)/(M*pi)*integral(fun,0,pi/2);
end
end
end

Output 

 



MATLAB Code for flat fading channel


% OFDM over flat Rayleigh fading channel
clc; clearvars; close all;
% Simulation parameters
nSym = 10^4; % Number of OFDM symbols
EbN0dB = 0:2:20; % Eb/N0 range
MOD_TYPE = 'MPSK'; % 'MPSK' or 'MQAM'
M = 4; % QPSK
N = 64; % Total number of subcarriers
Ncp = 16; % Cyclic prefix length
k = log2(M); % Bits per symbol
EsN0dB = 10*log10(k*(N/(N+Ncp))) + EbN0dB; % account CP loss
errors = zeros(1,length(EsN0dB));
% Monte Carlo simulation
for i = 1:length(EsN0dB)
for j = 1:nSym
% -------- Transmitter --------
d = ceil(M*rand(1,N)); % Random symbols
[X , ref] = modulation_mapper(MOD_TYPE,M,d);
x = ifft(X,N); % IDFT
s = add_cyclic_prefix(x,Ncp); % Add CP
% -------- Channel (Flat Rayleigh) --------
h = 1/sqrt(2)*(randn + 1i*randn); % Single-tap flat fading
H = h; % Frequency response
r = h*s; % Flat-fading multiplication
r = awgn(r,EsN0dB(i),'measured'); % Add AWGN noise
% -------- Receiver --------
y = remove_cyclic_prefix(r,Ncp,N); % Remove CP
Y = fft(y,N); % DFT
V = Y ./ H; % Equalization
[~, dcap] = iqOptDetector(V,ref);
errors(i) = errors(i) + sum(d ~= dcap);
end
end
% SER results
simulatedSER = errors/(nSym*N);
theoreticalSER = ser_rayleigh(EbN0dB,MOD_TYPE,M);
% Plot
semilogy(EbN0dB,simulatedSER,'ko'); hold on;
title(['Performance of ',num2str(M),'-',MOD_TYPE,...
' OFDM over Flat Rayleigh channel']);
xlabel('Eb/N0 (dB)');
ylabel('Symbol Error Rate');
%% ------------------------ Functions ------------------------
function [X, ref] = modulation_mapper(MOD_TYPE, M, d)
if strcmpi(MOD_TYPE, 'MPSK')
[X, ref] = mpsk_modulator(M, d);
elseif strcmpi(MOD_TYPE, 'MQAM')
[X, ref] = mqam_modulator(M, d);
else
error('Invalid modulation type. Use ''MPSK'' or ''MQAM''.');
end
end
function [s, ref] = mpsk_modulator(M,d)
ref_i= 1/sqrt(2)*cos(((1:M)-1)/M*2*pi);
ref_q= 1/sqrt(2)*sin(((1:M)-1)/M*2*pi);
ref = ref_i + 1i*ref_q;
s = ref(d);
end
function s = add_cyclic_prefix(x,Ncp)
s = [x(end-Ncp+1:end) x];
end
function y = remove_cyclic_prefix(r,Ncp,N)
y = r(Ncp+1:N+Ncp);
end
function [idealPoints,indices] = iqOptDetector(received,ref)
x = [real(received); imag(received)]';
y = [real(ref); imag(ref)]';
[idealPoints,indices] = minEuclideanDistance(x,y);
end
function [idealPoints,indices] = minEuclideanDistance(x,y)
[m,p1] = size(x); [n,p2] = size(y);
if p1 ~= p2, error('Dimension mismatch'); end
X = sum(x.*x,2);
Y = sum(y.*y,2)';
d = X(:,ones(1,n)) + Y(ones(1,m),:) - 2*x*y';
[~,indices] = min(d,[],2);
idealPoints = y(indices,:);
indices = indices.';
end
function ser = ser_rayleigh(EbN0dB,MOD_TYPE,M)
gamma_b = 10.^(EbN0dB/10); % SNR per bit linear
gamma_s = log2(M)*gamma_b; % SNR per symbol linear
switch lower(MOD_TYPE)
case {'bpsk'}
ser = 0.5*(1-sqrt(gamma_b./(1+gamma_b)));
case {'mpsk','psk'}
ser = zeros(size(gamma_s));
for i=1:length(gamma_s)
g = sin(pi/M)^2;
fun = @(x) 1./(1 + (g*gamma_s(i)./(sin(x).^2)));
ser(i) = (1/pi)*integral(fun,0,pi*(M-1)/M);
end
case {'mqam','qam'}
ser = zeros(size(gamma_s));
for i=1:length(gamma_s)
g = 1.5/(M-1);
fun = @(x) 1./(1 + (g*gamma_s(i)./(sin(x).^2)));
ser(i) = 4/pi*(1-1/sqrt(M))*integral(fun,0,pi/2) ...
- 4/pi*(1-1/sqrt(M))^2*integral(fun,0,pi/4);
end
end
end
 

 

 Output

 

 


Further Reading


📊 MATLAB & ONLINE SIMULATIONS (Web-Based)
🎓 GATE & UGC - NET SOLUTIONS
📡 WIRELESS & SIGNAL PROCESSING


Contact Us

Name

Email *

Message *

Popular Posts

UGC NET Electronic Science Previous Year Question Papers with Solutions

Home / Engineering & Other Exams / UGC NET 2026 PYQ ⬇️ Download Papers and Solutions 📋 Exam Pattern 💡 Preparation Tips ❓ FAQs 📊 Exam Highlights: Electronic Science (88) Feature Details Junior Research Fellowship (JRF) ₹37,000 + HRA per month Eligibility M.Sc/M.Tech in Electronics (55%) Validity of Certificate JRF (3 Years) | Lectureship (Lifetime) 📥 Download UGC NET Electronics PDFs Complete collection of previous year question papers, answer keys and explanations for Subject Code 88. Start Downloading 📂 View All Question Papers June 2025 - Question Paper Download PDF June 2025 - Solved Paper + Explanation ...
Read more

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...
Read more

Constellation Diagrams of ASK, PSK, and FSK (with MATLAB Code + Simulator)

Constellation Diagrams: ASK, FSK, and PSK Comprehensive guide to signal space representation, including interactive simulators and MATLAB implementations. 📘 Overview 🧮 Simulator ⚖️ Theory 📈 Q-function 📚 Resources BASK Modulation Transmits one of two signals: 0 or $\sqrt{E_b}$, representing binary 0 and 1. Simple but sensitive to noise. BFSK Modulation Transmits one of two signals: $\sqrt{E_b}$ on the Y-axis or $\sqrt{E_b}$ on the X-axis. These are orthogonal signals. BPSK Modulation Transmits $+\sqrt{E_b}$ or $-\sqrt{E_b}$ (antipodal signaling). Most efficient binary scheme. ...
Read more

Wiener Filter in MATLAB

  MATLAB Code  % Wiener Filter Based on Wiener-Hopf Equation % This script demonstrates how to apply the Wiener filter to recover % a reference signal from a noisy signal using the Wiener-Hopf equation. % The filter minimizes the mean squared error between the noisy signal and the reference signal. clear; close all; clc; % Signal Parameters fs = 4000; % Sampling frequency (Hz) T = 1; % Total recording time (seconds) L = T * fs; % Signal length (samples) tt = (0:L-1) / fs; % Time vector ff = (0:L-1) * fs / L; % Frequency vector % Generate Reference Signal (a sinusoid) y = sin(2 * pi * 120 * tt); % Reference sinusoidal signal y = y(:); % Ensure column vector % Create Noisy Signal by Adding Gaussian Noise x = 0.50 * randn(L, 1) + y; % Noisy signal x = x(:); % Ensure column vector % Define Filter Order (Number of Coefficients) N = 200; % Apply Wiener Filter using custom function [xest, b, MSE] = wienerFilt(x, y, N); % Plot Results figure; subplot(411); plot(tt, x, 'k'), hold on, p...
Read more

MIMO Channel Matrix | Rank and Condition Number

MIMO / Massive MIMO MIMO Channel Matrix | Rank and Condition...   The channel matrix in wireless communication is a matrix that describes the impact of the channel on the transmitted signal. The channel matrix can be used to model the effects of the atmospheric or underwater environment on the signal, such as the absorption, reflection or scattering of the signal by surrounding objects. When addressing multi-antenna communication, the term "channel matrix" is used. Let's assume that only one TX and one RX are in communication and there's no surrounding object. Here, in our case, we can apply the proper threshold condition to a received signal and get the original transmitted signal at the RX side. However, in real-world situations, we see signal path blockage, reflections, etc.,  (NLOS paths [↗]) more frequently. The obstruction is typically caused by building walls, etc. Multi-antenna communication was introduced to address this issue. It makes diversity app...
Read more

Overmodulation & Distortion in AM

Overmodulation in AM and How It Causes Distortion 1. AM Signal Equation s(t) = A c [1 + μ m(t)] cos(2Ï€ f c t) A c = carrier amplitude m(t) = normalized modulating signal (|m(t)| ≤ 1) μ = modulation index 2. Modulation Index μ = A m / A c - Normal AM: 0 < μ ≤ 1 → no distortion - Overmodulation: μ > 1 → distortion occurs 3. Envelope and Overmodulation A(t) = A c [1 + μ m(t)] - For undistorted AM: 1 + μ m(t) ≥ 0 at all times - If μ > 1: 1 + μ m(t) < 0 at negative peaks → carrier flips Example: Let m(t) = cos(2Ï€ f m t), A c = 1 V, μ = 1.2 Minimum envelope: A min = A c [1 - 1.2] = -0.2 V Negative amplitude → envelope crosses zero → 180° phase flip 4. Mathematical Consequence -A c cos(θ) = A c cos(θ + Ï€) This phase reversal is what causes distortion in the demodulated signal. 5. Instantaneous AM Signal s...
Read more

MUSIC Algorithm Explained (with MATLAB + Simulator)

Practical Implementation of the MUSIC Algorithm The focus is on how the algorithm works computationally , not just theory, and it explains the denominator (a H E n E n H a) mathematically and intuitively. 1. Introduction The MUSIC (Multiple Signal Classification) algorithm is a high-resolution method used in signal processing and array processing to estimate the Direction of Arrival (DOA) of signals received by a sensor array. Unlike classical beamforming methods, MUSIC uses eigenvector decomposition of the covariance matrix to separate the signal subspace and noise subspace , allowing it to achieve much higher angular resolution. In practical implementations, MUSIC works by: Simulating or collecting array signals Computing the covariance matrix Performing eigenvalue decomposition Separating signal and noise subspaces Scanning possible angles using a steering vector Constructing a pseudo-spectrum where peaks indicate signal directions 2. Signal Mo...
Read more

OFDM Symbols and Subcarriers Explained

This article explains how OFDM (Orthogonal Frequency Division Multiplexing) symbols and subcarriers work. It covers modulation, mapping symbols to subcarriers, subcarrier frequency spacing, IFFT synthesis, cyclic prefix, and transmission. Step 1: Modulation First, modulate the input bitstream. For example, with 16-QAM , each group of 4 bits maps to one QAM symbol. Suppose we generate a sequence of QAM symbols: s0, s1, s2, s3, s4, s5, …, s63 Step 2: Mapping Symbols to Subcarriers Assume N sub = 8 subcarriers. Each OFDM symbol in the frequency domain contains 8 QAM symbols (one per subcarrier): Mapping (example) OFDM symbol 1 → s0, s1, s2, s3, s4, s5, s6, s7 OFDM symbol 2 → s8, s9, s10, s11, s12, s13, s14, s15 … OFDM sym...
Read more