Skip to main content

Periodogram in MATLAB


Step 1: Signal Representation

Let the signal be x[n], where:

  • n = 0, 1, ..., N-1 (discrete-time indices),
  • N is the total number of samples.

Step 2: Compute the Discrete-Time Fourier Transform (DTFT)

The DTFT of x[n] is:

X(f) = ∑ x[n] e-j2Ï€fn

For practical computation, the Discrete Fourier Transform (DFT) is used:

X[k] = ∑ x[n] e-j(2Ï€/N)kn, k = 0, 1, ..., N-1

Here:

  • k represents discrete frequency bins,
  • f_k = k/N * f_s, where f_s is the sampling frequency.

Step 3: Compute Power Spectral Density (PSD)

The periodogram estimates the PSD as:

S_x(f_k) = (1/N) |X[k]|²

Where:

  • S_x(f_k) represents the power of the signal at frequency f_k.
  • The factor 1/N normalizes the power by the signal length.

Step 4: Convert to Decibels (Optional)

For visualization, convert PSD to decibels (dB):

S_xdB(f_k) = 10 log₁₀(S_x(f_k))

Practical Notes

  • Frequency Resolution: Depends on the signal duration T = N / f_s. Higher N gives finer frequency resolution.
  • Windowing (Optional): Use a window function (e.g., Hamming, Hann) to reduce spectral leakage: x'[n] = x[n] * w[n].
  • Frequency Range: PSD spans frequencies:
    • f_k = k/N * f_s for positive frequencies.
    • f_k = -(N-k)/N * f_s for negative frequencies.

     

    MATLAB Code

    clc;
    clear;
    close all;

    % Define signal parameters
    N = 256; % Number of samples
    fs = 1000; % Sampling frequency in Hz
    t = (0:N-1)/fs; % Time vector

    % Generate a sample signal (sum of two sinusoids)
    f1 = 50; % Frequency of first sinusoid in Hz
    f2 = 120; % Frequency of second sinusoid in Hz
    x = sin(2*pi*f1*t) + 0.5*sin(2*pi*f2*t) + randn(1, N)*0.1; % Signal with noise

    % Compute DFT of the signal
    X = fft(x, N);

    % Calculate the periodogram
    Pxx = (1/N) * abs(X).^2;

    % Generate frequency vector
    f = (0:N-1)*(fs/N);

    % Keep only the positive frequencies (up to Nyquist frequency)
    Pxx = Pxx(1:N/2+1);
    f = f(1:N/2+1);

    % Plot the periodogram
    figure;
    plot(f, 10*log10(Pxx), 'LineWidth', 1.5);
    xlabel('Frequency (Hz)');
    ylabel('Power/Frequency (dB/Hz)');
    title('Periodogram');
    grid on;

    Output 

     


     

     

     

     

    Copy the MATLAB Code above from here

     

    Further Reading

    1. Periodogram and Windowed Periododgram in details
    2. Correlogram in MATLAB
    3. Bartlett Method in MATLAB
    4. Blackman-Tukey Method in MATLAB
    5. Welch's Method in MATLAB

People are good at skipping over material they already know!

View Related Topics to







Admin & Author: Salim

s

  Website: www.salimwireless.com
  Interests: Signal Processing, Telecommunication, 5G Technology, Present & Future Wireless Technologies, Digital Signal Processing, Computer Networks, Millimeter Wave Band Channel, Web Development
  Seeking an opportunity in the Teaching or Electronics & Telecommunication domains.
  Possess M.Tech in Electronic Communication Systems.


Contact Us

Name

Email *

Message *

Popular Posts

BER vs SNR for M-ary QAM, M-ary PSK, QPSK, BPSK, ...

📘 Overview of BER and SNR 🧮 Simulator for m-ary QAM and m-ary PSK 🧮 MATLAB Codes 📚 Further Reading Modulation Constellation Diagrams BER vs. SNR BER vs SNR for M-QAM, M-PSK, QPSk, BPSK, ... What is Bit Error Rate (BER)? The abbreviation BER stands for bit error rate, which indicates how many corrupted bits are received (after the demodulation process) compared to the total number of bits sent in a communication process. It is defined as,  In mathematics, BER = (number of bits received in error / total number of transmitted bits)  On the other hand, SNR refers to the signal-to-noise power ratio. For ease of calculation, we commonly convert it to dB or decibels.   What is Signal the signal-to-noise ratio (SNR)? SNR = signal power/noise power (SNR is a ratio of signal power to noise power) SNR (in dB) = 10*log(signal power / noise power) [base 10] For instance,...

UGC NET Electronic Science Previous Year Question Papers

Home / Engineering & Other Exams / UGC NET 2022: Previous Year Question Papers ...   NET | GATE | ESE | UGC-NET (Electronics Science, Subject code: 88 ) UGC Net Electronic Science Questions Paper With Answer Key Download Pdf [December 2024] UGC Net Electronic Science Questions Paper With Answer Key Download Pdf [June 2024] UGC Net Electronic Science Questions Paper With Answer Key Download Pdf [December 2023] UGC Net Electronic Science Questions Paper With Answer Key Download Pdf [June 2023] UGC Net Electronic Science Questions Paper With Answer Key Download Pdf [December 2022]  UGC Net Electronic Science Questions Paper With Answer Key Download Pdf [June 2022]   UGC Net Electronic Science Questions Paper With Answer Key Download Pdf [December 2021] UGC Net Electronic Science Questions With Answer Key Download Pdf [June 2020] UGC Net Electronic Science Questions With Answer Key Download Pdf [December 2019] UGC Net Electronic Science Questions With Answer...

Constellation Diagrams of ASK, PSK, and FSK

📘 Overview 🧮 Simulator for constellation diagrams of ASK, FSK, and PSK 🧮 Theory 🧮 MATLAB Codes 🧮 Simulator for constellation diagrams of m-ary PSK 🧮 Simulator for constellation diagrams of m-ary QAM 📚 Further Reading BASK (Binary ASK) Modulation: Transmits one of two signals: 0 or -√Eb, where Eb​ is the energy per bit. These signals represent binary 0 and 1.    BFSK (Binary FSK) Modulation: Transmits one of two signals: +√Eb​ ( On the y-axis, the phase shift of 90 degrees with respect to the x-axis, which is also termed phase offset ) or √Eb (on x-axis), where Eb​ is the energy per bit. These signals represent binary 0 and 1.  BPSK (Binary PSK) Modulation: Transmits one of two signals: +√Eb​ or -√Eb (they differ by 180 degree phase shift), where Eb​ is the energy per bit. These signals represent binary 0 and 1.    Simulator for BASK, BPSK, and BFSK Constellation Diagrams ...

MATLAB Code for Pulse Amplitude Modulation (PAM) and Demodulation

📘 Overview & Theory 🧮 MATLAB Code 1 🧮 MATLAB Code 2 🧮 MATLAB Code for Pulse Amplitude Modulation and Demodulation of Digital data 🧮 Other Pulse Modulation Techniques (e.g., PWM, PPM, DM, and PCM) 📚 Further Reading   Pulse Amplitude Modulation (PAM) & Demodulation MATLAB Script clc; clear all; close all; fm= 10; % frequency of the message signal fc= 100; % frequency of the carrier signal fs=1000*fm; % (=100KHz) sampling frequency (where 1000 is the upsampling factor) t=0:1/fs:1; % sampling rate of (1/fs = 100 kHz) m=1*cos(2*pi*fm*t); % Message signal with period 2*pi*fm (sinusoidal wave signal) c=0.5*square(2*pi*fc*t)+0.5; % square wave with period 2*pi*fc s=m.*c; % modulated signal (multiplication of element by element) subplot(4,1,1); plot(t,m); title('Message signal'); xlabel ('Time'); ylabel('Amplitude'); subplot(4,1,2); plot(t,c); title('Carrier signal'); xlabel('Time'); ylabel('Amplitu...

Alamouti Scheme for 2x2 MIMO in MATLAB

📘 Overview & Theory 🧮 MATLAB Code for Alamouti Scheme 🧮 MATLAB Code for BER vs. SNR for Alamouti Scheme 🧮 Alamouti Scheme Transmission Table 📚 Further Reading    Read about the Alamouti Scheme first MATLAB Code for Alamouti's Precoding Matrix for 2 X 2 MIMO % Clear any existing data and figures clc; clear; close all; % Define system parameters transmitAntennas = 2; % Number of antennas at the transmitter receiveAntennas = 2; % Number of antennas at the receiver symbolCount = 1000000; % Number of symbols to transmit SNR_dB = 15; % Signal-to-Noise Ratio in decibels % Generate random binary data for transmission rng(10); % Set seed for reproducibility transmitData = randi([0, 1], transmitAntennas, symbolCount); % Perform Binary Phase Shift Keying (BPSK) modulation modulatedSymbols = 1 - 2 * transmitData; % Define Alamouti's Precoding Matrix precodingMatrix = [1 1; -1i 1i]; % Encode and transmit data using Alamouti scheme transmittedSym...

Comparisons among ASK, PSK, and FSK | And the definitions of each

📘 Overview 🧮 Simulator 🧮 Noise Sensitivity, Bandwidth, Complexity, etc. 🧮 MATLAB Codes 🧮 Some Questions and Answers 📚 Further Reading Modulation ASK, FSK & PSK Constellation MATLAB Simulink MATLAB Code Comparisons among ASK, PSK, and FSK    Comparisons among ASK, PSK, and FSK   Simulator for Calculating Bandwidth of ASK, FSK, and PSK The baud rate represents the number of symbols transmitted per second. Both baud rate and bit rate are same for binary ASK, FSK, and PSK. Select Modulation Type: ASK FSK PSK Baud Rate or Bit Rate (bps): Frequency Deviation (Hz) for FSK: Calculate Bandwidth Comparison among ASK,  FSK, and PSK Performance Comparison: 1. Noise Sensitivity:    - ASK is the most sensitive to noise due to its r...

Theoretical vs. simulated BER vs. SNR for ASK, FSK, and PSK

📘 Overview 🧮 Simulator for calculating BER 🧮 MATLAB Codes for calculating theoretical BER 🧮 MATLAB Codes for calculating simulated BER 📚 Further Reading   BER vs. SNR denotes how many bits in error are received in a communication process for a particular Signal-to-noise (SNR) ratio. In most cases, SNR is measured in decibel (dB). For a typical communication system, a signal is often affected by two types of noises 1. Additive White Gaussian Noise (AWGN) 2. Rayleigh Fading In the case of additive white Gaussian noise (AWGN), random magnitude is added to the transmitted signal. On the other hand, Rayleigh fading (due to multipath) attenuates the different frequency components of a signal differently. A good signal-to-noise ratio tries to mitigate the effect of noise.  Simulator for calculating BER vs SNR for binary ASK, FSK, and PSK Calculate BER for Binary ASK Modulation The theoretical BER for binary ASK (BASK) in an AWGN channel is...

MATLAB code for BER vs SNR for M-QAM, M-PSK, QPSk, BPSK, ...

📘 Overview 🧮 Simulator for m-ary QAM and m-ary PSK 🧮 MATLAB Code for BPSK, M-ary PSK, and M-ary QAM Together 🧮 MATLAB Code for M-ary QAM 🧮 MATLAB Code for M-ary PSK 📚 Further Reading   MATLAB Script for  BER vs. SNR for M-QAM, M-PSK, QPSk, BPSK %Written by Salim Wireless %Visit www.salimwireless.com for study materials on wireless communication %or, if you want to learn how to code in MATLAB clc; clear; close all; % Parameters num_symbols = 1e5; % Number of symbols snr_db = -20:2:20; % Range of SNR values in dB % PSK and QAM orders to be tested psk_orders = [2, 4, 8, 16, 32]; qam_orders = [4, 16, 64, 256]; % Initialize BER arrays ber_psk_results = zeros(length(psk_orders), length(snr_db)); ber_qam_results = zeros(length(qam_orders), length(snr_db)); % BER calculation for each PSK order and SNR value for i = 1:length(psk_orders) psk_order = psk_orders(i); for j = 1:length(snr_db) % Generate random symbols ...