FFT and frequency analysis in MATLAB

Introduction

FFT helps us find hidden repeating patterns or frequencies in data. It turns time-based data into frequency-based data so we can understand signals better.

To find the main tones in a sound recording.

To analyze vibrations in a machine to detect problems.

To study heartbeats or brain waves in medical data.

To check the quality of a signal in communications.

To identify repeating cycles in stock market prices.

Syntax

MATLAB

Y = fft(X)

% X is your input data (like a signal)
% Y is the output with frequency information

The input X can be a vector or matrix of numbers.

The output Y is complex numbers showing amplitude and phase of frequencies.

Examples

Computes FFT of a simple 4-point signal.

MATLAB

Y = fft([1 2 3 4])

Computes FFT with zero-padding to length 1024 for better frequency resolution.

MATLAB

Y = fft(signal, 1024)

Calculates the single-sided amplitude spectrum from the FFT output.

MATLAB

P2 = abs(Y/length(Y))
P1 = P2(1:length(Y)/2+1)
P1(2:end-1) = 2*P1(2:end-1);

Sample Program

This program creates a noisy signal with two known frequencies. It uses FFT to find the frequency components and plots the amplitude spectrum showing peaks at those frequencies.

MATLAB

% Sample signal: sum of two sine waves
Fs = 1000;            % Sampling frequency in Hz
T = 1/Fs;             % Sampling period
L = 1500;             % Length of signal
t = (0:L-1)*T;        % Time vector

% Create a signal with two frequencies 50 Hz and 120 Hz
S = 0.7*sin(2*pi*50*t) + sin(2*pi*120*t);

% Add some noise
X = S + 2*randn(size(t));

% Compute FFT
Y = fft(X);

% Compute two-sided spectrum P2
P2 = abs(Y/L);

% Compute single-sided spectrum P1 based on P2 and even-valued L
P1 = P2(1:L/2+1);
P1(2:end-1) = 2*P1(2:end-1);

% Define frequency domain f
f = Fs*(0:(L/2))/L;

% Plot single-sided amplitude spectrum
plot(f,P1)
title('Single-Sided Amplitude Spectrum of X(t)')
xlabel('Frequency (Hz)')
ylabel('|P1(f)|')

Important Notes

FFT output is complex; use abs() to get amplitude.

Frequency resolution depends on sampling rate and signal length.

Plotting the single-sided spectrum helps see positive frequencies clearly.

Summary

FFT converts time signals into frequency components.

Use fft() in MATLAB to perform frequency analysis.

Amplitude spectrum shows strength of each frequency in the signal.

Practice

1. What does the MATLAB function fft() do to a signal?

easy

A. It amplifies the signal amplitude.

B. It converts the signal from time domain to frequency domain.

C. It smooths the signal to remove noise.

D. It shifts the signal in time.

Solution

1. Step 1: Understand the purpose of FFT

   FFT stands for Fast Fourier Transform, which changes a signal from time domain to frequency domain.

2. Step 2: Identify what fft() does in MATLAB

   In MATLAB, fft() computes the discrete Fourier transform, showing frequency components of the signal.

3. Final Answer:

   It converts the signal from time domain to frequency domain. -> Option B

4. Quick Check:

   FFT = frequency transform [OK]

Hint: FFT always changes time signals to frequency signals [OK]

Common Mistakes:

• Thinking FFT smooths or filters signals
• Confusing FFT with time shifting
• Assuming FFT amplifies signal amplitude

2. Which of the following is the correct syntax to compute the FFT of a vector x in MATLAB?

easy

A. Y = fft x;

B. Y = fft[x];

C. Y = fft(x);

D. Y = fft{x};

Solution

1. Step 1: Recall MATLAB function call syntax

   In MATLAB, functions are called with parentheses enclosing arguments, e.g., func(arg).

2. Step 2: Check each option's syntax

   Y = fft(x); uses correct syntax: fft(x). Options A, C, and D use invalid syntax for function calls in MATLAB.

3. Final Answer:

   Y = fft(x); -> Option C

4. Quick Check:

   Function calls use parentheses () [OK]

Hint: Use parentheses () for function calls in MATLAB [OK]

Common Mistakes:

• Using square brackets [] instead of parentheses
• Omitting parentheses around arguments
• Using curly braces {} which are for cell arrays

3. Given the MATLAB code:

x = [1 1 0 0];
Y = fft(x);
absY = abs(Y);
disp(absY);

What is the output displayed?

medium

A. [1 1 1 1]

B. [2 0 2 0]

C. [4 0 0 0]

D. [2 1.4142 0 1.4142]

Solution

1. Step 1: Compute FFT of x = [1 1 0 0]

   The FFT of this sequence results in complex values whose magnitudes are [2, sqrt(2), 0, sqrt(2)] approximately.

2. Step 2: Calculate absolute values

   Taking absolute values gives approximately [2, 1.4142, 0, 1.4142].

3. Final Answer:

   [2 1.4142 0 1.4142] -> Option D

4. Quick Check:

   FFT magnitude matches [2 1.4142 0 1.4142] [OK]

Hint: FFT magnitude shows signal strength at each frequency [OK]

Common Mistakes:

• Confusing FFT output with original signal
• Not taking absolute value before display
• Assuming FFT output is all real numbers

4. Consider this MATLAB code snippet:

Fs = 1000;
t = 0:1/Fs:1-1/Fs;
x = cos(2*pi*50*t) + cos(2*pi*120*t);
Y = fft(x);
P2 = abs(Y/length(x));
P1 = P2(1:length(x)/2+1);
P1(2:end-1) = 2*P1(2:end-1);
plot(P1);

What is the likely error in this code?

medium

A. The frequency axis is missing for the plot, so frequencies are unclear.

B. The length of x is not used correctly in dividing Y.

C. The FFT function is called incorrectly without arguments.

D. The signal x is not defined properly.

Solution

1. Step 1: Analyze the FFT and amplitude spectrum calculation

   The code correctly computes the single-sided amplitude spectrum P1 from Y.

2. Step 2: Check the plot command

   The plot shows amplitude values but does not label or define the frequency axis, making interpretation difficult.

3. Final Answer:

   The frequency axis is missing for the plot, so frequencies are unclear. -> Option A

4. Quick Check:

   Plot frequency axis needed for clarity [OK]

Hint: Always plot frequency axis with amplitude spectrum [OK]

Common Mistakes:

• Forgetting to create frequency vector for x-axis
• Plotting amplitude without frequency labels
• Misusing FFT input or output sizes

5. You have a noisy signal sampled at 1000 Hz stored in vector data. You want to find the dominant frequency component using FFT. Which MATLAB code snippet correctly computes and plots the single-sided amplitude spectrum with frequency axis?

hard

A. L = length(data); Fs = 1000; Y = fft(data); P2 = abs(Y/L); f = Fs*(0:(L/2))/L; P1 = P2(1:L/2+1); P1(2:end-1) = 2*P1(2:end-1); plot(f,P1); xlabel('Frequency (Hz)'); ylabel('Amplitude');

B. Fs = 1000; Y = fft(data); plot(abs(Y)); xlabel('Frequency'); ylabel('Amplitude');

C. L = length(data); Y = fft(data); P1 = abs(Y); plot(P1);

D. Fs = 1000; t = 0:1/Fs:1; Y = fft(data); plot(Y);

Solution

1. Step 1: Understand the steps to compute single-sided amplitude spectrum

   Calculate FFT, normalize by length, create frequency vector, extract half spectrum, double amplitudes except DC and Nyquist.

2. Step 2: Check which option follows these steps correctly

   L = length(data); Fs = 1000; Y = fft(data); P2 = abs(Y/L); f = Fs*(0:(L/2))/L; P1 = P2(1:L/2+1); P1(2:end-1) = 2*P1(2:end-1); plot(f,P1); xlabel('Frequency (Hz)'); ylabel('Amplitude'); correctly computes length, frequency vector, normalizes, adjusts amplitudes, and plots with labels. Others miss frequency axis or normalization.

3. Final Answer:

   Option A code snippet -> Option A

4. Quick Check:

   Proper FFT amplitude spectrum needs frequency axis and normalization [OK]

Hint: Normalize FFT and plot with frequency vector for correct spectrum [OK]

Common Mistakes:

• Not normalizing FFT by signal length
• Plotting full FFT without single-sided spectrum
• Missing frequency axis in plot