FFT and frequency analysis in MATLAB - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does FFT stand for in signal processing?

FFT stands for Fast Fourier Transform. It is an efficient way to compute the Discrete Fourier Transform (DFT) of a signal.

beginner

What is the main purpose of performing frequency analysis using FFT?

The main purpose is to find the different frequency components present in a time-domain signal, helping us understand its behavior in the frequency domain.

beginner

In MATLAB, which function is used to compute the FFT of a signal?

The function used is fft(). For example, Y = fft(x) computes the FFT of signal x.

intermediate

Why do we often use abs() on the FFT output in MATLAB?

Because the FFT output is complex, abs() gives the magnitude of each frequency component, which shows the strength of that frequency in the signal.

intermediate

What is the relationship between the sampling frequency and the frequency axis when plotting FFT results?

The sampling frequency determines the maximum frequency we can analyze (Nyquist frequency). The frequency axis is scaled from 0 to half the sampling frequency.

What does the FFT output represent?

A Amplitude of the original signal only

B Time delay of the signal

C Noise in the signal

D Frequency components of the input signal

FFT output shows the frequency components present in the input signal.

Which MATLAB function computes the Fast Fourier Transform?

A ifft()

B fft()

C filter()

D plot()

fft() computes the Fast Fourier Transform.

Why do we use abs() on FFT results?

A To get the magnitude of frequency components

B To get the phase of frequency components

C To convert to time domain

D To normalize the signal

abs() extracts the magnitude from complex FFT output.

What is the Nyquist frequency?

A The frequency with highest amplitude

B Twice the sampling frequency

C Half the sampling frequency

D Zero frequency

Nyquist frequency is half the sampling frequency and is the highest frequency that can be accurately represented.

If a signal is sampled at 1000 Hz, what is the maximum frequency shown in its FFT plot?

A 500 Hz

B 1000 Hz

C 2000 Hz

D 250 Hz

The maximum frequency is half the sampling rate, so 500 Hz.

Explain how FFT helps in analyzing a time-domain signal.

Think about how music can be broken down into notes.

Describe the steps to plot the magnitude spectrum of a signal using MATLAB's FFT.

Start from the signal and end with a graph.

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