fftshift Archives - GaussianWaves

Interpret FFT results – obtaining magnitude and phase information

In the previous post, Interpretation of frequency bins, frequency axis arrangement (fftshift/ifftshift) for complex DFT were discussed. In this post, I intend to show you how to interpret FFT results and obtain magnitude and phase information.

Outline

For the discussion here, lets take an arbitrary cosine function of the form $x(t)= A cos \left(2 \pi f_c t + \phi \right)$ and proceed step by step as follows

● Represent the signal $x(t)$ in computer (discrete-time) and plot the signal (time domain)
● Represent the signal in frequency domain using FFT ($ X[k]$)
● Extract amplitude and phase information from the FFT result
● Reconstruct the time domain signal from the frequency domain samples

This article is part of the book Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885 available in ebook (PDF) format (click here) and Paperback (hardcopy) format (click here)
Wireless Communication Systems in Matlab, ISBN: 978-1720114352 available in ebook (PDF) format (click here) and Paperback (hardcopy) format (click here).

Discrete-time domain representation

Consider a cosine signal of amplitude $A=0.5$, frequency $f_c=10 Hz$ and phase $phi= \pi/6$ radians (or $30^{\circ}$ )

\[x(t) = 0.5 cos \left( 2 \pi 10 t + \pi/6 \right)\]

In order to represent the continuous time signal $x(t)$ in computer memory, we need to sample the signal at sufficiently high rate (according to Nyquist sampling theorem). I have chosen a oversampling factor of $32$ so that the sampling frequency will be $f_s = 32 \times f_c $, and that gives $640$ samples in a $2$ seconds duration of the waveform record.

A = 0.5; %amplitude of the cosine wave
fc=10;%frequency of the cosine wave
phase=30; %desired phase shift of the cosine in degrees
fs=32*fc;%sampling frequency with oversampling factor 32
t=0:1/fs:2-1/fs;%2 seconds duration

phi = phase*pi/180; %convert phase shift in degrees in radians
x=A*cos(2*pi*fc*t+phi);%time domain signal with phase shift

figure; plot(t,x); %plot the signal

Represent the signal in frequency domain using FFT

Lets represent the signal in frequency domain using the FFT function. The FFT function computes $N$-point complex DFT. The length of the transformation $N$ should cover the signal of interest otherwise we will some loose valuable information in the conversion process to frequency domain. However, we can choose a reasonable length if we know about the nature of the signal.

For example, the cosine signal of our interest is periodic in nature and is of length $640$ samples (for 2 seconds duration signal). We can simply use a lower number $N=256$ for computing the FFT. In this case, only the first $256$ time domain samples will be considered for taking FFT. No need to worry about loss of information in this case, as the $256$ samples will have sufficient number of cycles using which we can calculate the frequency information.

N=256; %FFT size
X = 1/N*fftshift(fft(x,N));%N-point complex DFT

In the code above, $fftshift$ is used only for obtaining a nice double-sided frequency spectrum that delineates negative frequencies and positive frequencies in order. This transformation is not necessary. A scaling factor $1/N$ was used to account for the difference between the FFT implementation in Matlab and the text definition of complex DFT.

3a. Extract amplitude of frequency components (amplitude spectrum)

The FFT function computes the complex DFT and the hence the results in a sequence of complex numbers of form $X_{re} + j X_{im}$. The amplitude spectrum is obtained

\[|X[k]| = \sqrt{X_{re}^2 + X_{im}^2 } \]

For obtaining a double-sided plot, the ordered frequency axis (result of fftshift) is computed based on the sampling frequency and the amplitude spectrum is plotted.

df=fs/N; %frequency resolution
sampleIndex = -N/2:N/2-1; %ordered index for FFT plot
f=sampleIndex*df; %x-axis index converted to ordered frequencies
stem(f,abs(X)); %magnitudes vs frequencies
xlabel('f (Hz)'); ylabel('|X(k)|');

3b. Extract phase of frequency components (phase spectrum)

Extracting the correct phase spectrum is a tricky business. I will show you why it is so. The phase of the spectral components are computed as

\[\angle X[k] = tan^{-1} \left( \frac{X_{im}}{X_{re}} \right)\]

That equation looks naive, but one should be careful when computing the inverse tangents using computers. The obvious choice for implementation seems to be the $atan$ function in Matlab. However, usage of $atan$ function will prove disastrous unless additional precautions are taken. The $atan$ function computes the inverse tangent over two quadrants only, i.e, it will return values only in the $[-\pi/2 , \pi/2]$ interval. Therefore, the phase need to be unwrapped properly. We can simply fix this issue by computing the inverse tangent over all the four quadrants using the $atan2(X_{img},X_{re})$ function.

Lets compute and plot the phase information using $atan2$ function and see how the phase spectrum looks

phase=atan2(imag(X),real(X))*180/pi; %phase information
plot(f,phase); %phase vs frequencies

The phase spectrum is completely noisy. Unexpected !!!. The phase spectrum is noisy due to fact that the inverse tangents are computed from the $ratio$ of imaginary part to real part of the FFT result. Even a small floating rounding off error will amplify the result and manifest incorrectly as useful phase information (read how a computer program approximates very small numbers).

To understand, print the first few samples from the FFT result and observe that they are not absolute zeros (they are very small numbers in the order $10^{-16}$. Computing inverse tangent will result in incorrect results.

>> X(1:5)
ans =
   1.0e-16 *
  -0.7286            -0.3637 - 0.2501i  -0.4809 - 0.1579i  -0.3602 - 0.5579i   0.0261 - 0.4950i
>> atan2(imag(X(1:5)),real(X(1:5)))
ans =
    3.1416   -2.5391   -2.8244   -2.1441   -1.5181

The solution is to define a tolerance threshold and ignore all the computed phase values that are below the threshold.

X2=X;%store the FFT results in another array
%detect noise (very small numbers (eps)) and ignore them
threshold = max(abs(X))/10000; %tolerance threshold
X2(abs(X)<threshold) = 0; %maskout values that are below the threshold
phase=atan2(imag(X2),real(X2))*180/pi; %phase information
plot(f,phase); %phase vs frequencies

The recomputed phase spectrum is plotted below. The phase spectrum has correctly registered the $30^{\circ}$ phase shift at the frequency $f=10 Hz$. The phase spectrum is anti-symmetric ($\phi=-30^{\circ}$ at $f=-10 Hz$ ), which is expected for real-valued signals.

Reconstruct the time domain signal from the frequency domain samples

Reconstruction of the time domain signal from the frequency domain sample is pretty straightforward

x_recon = N*ifft(ifftshift(X),N); %reconstructed signal
t = [0:1:length(x_recon)-1]/fs; %recompute time index 
plot(t,x_recon);%reconstructed signal

The reconstructed signal has preserved the same initial phase shift and the frequency of the original signal. Note: The length of the reconstructed signal is only $256$ sample long ($\approx 0.8$ seconds duration), this is because the size of FFT is considered as $N=256$. Since the signal is periodic it is not a concern. For more complicated signals, appropriate FFT length (better to use a value that is larger than the length of the signal) need to be used.

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Topics in this chapter

Essentials of Signal Processing
● Generating standard test signals
□ Sinusoidal signals
□ Square wave
□ Rectangular pulse
□ Gaussian pulse
□ Chirp signal
● Interpreting FFT results - complex DFT, frequency bins and FFTShift
□ Real and complex DFT
□ Fast Fourier Transform (FFT)
□ Interpreting the FFT results
□ FFTShift
□ IFFTShift
● Obtaining magnitude and phase information from FFT
□ Discrete-time domain representation
□ Representing the signal in frequency domain using FFT
□ Reconstructing the time domain signal from the frequency domain samples
● Power spectral density
● Power and energy of a signal
□ Energy of a signal
□ Power of a signal
□ Classification of signals
□ Computation of power of a signal - simulation and verification
● Polynomials, convolution and Toeplitz matrices
□ Polynomial functions
□ Representing single variable polynomial functions
□ Multiplication of polynomials and linear convolution
□ Toeplitz matrix and convolution
● Methods to compute convolution
□ Method 1: Brute-force method
□ Method 2: Using Toeplitz matrix
□ Method 3: Using FFT to compute convolution
□ Miscellaneous methods
● Analytic signal and its applications
□ Analytic signal and Fourier transform
□ Extracting instantaneous amplitude, phase, frequency
□ Phase demodulation using Hilbert transform
● Choosing a filter : FIR or IIR : understanding the design perspective
□ Design specification
□ General considerations in design

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Interpret FFT, complex DFT, frequency bins & FFTShift

Key focus: Interpret FFT results, complex DFT, frequency bins, fftshift and ifftshift. Know how to use them in analysis using Matlab and Python.

This article is part of the following books
● Digital Modulations using Matlab : Build Simulation Models from Scratch, ISBN: 978-1521493885
● Digital Modulations using Python ISBN: 978-1712321638
● Wireless communication systems in Matlab ISBN: 979-8648350779
All books available in ebook (PDF) and Paperback formats

Four types of Fourier Transforms:

Often, one is confronted with the problem of converting a time domain signal to frequency domain and vice-versa. Fourier Transform is an excellent tool to achieve this conversion and is ubiquitously used in many applications. In signal processing, a time domain signal can be continuous or discrete and it can be aperiodic or periodic. This gives rise to four types of Fourier transforms.

From Table 1, we note note that when the signal is discrete in one domain, it will be periodic in other domain. Similarly, if the signal is continuous in one domain, it will be aperiodic (non-periodic) in another domain. For simplicity, let’s not venture into the specific equations for each of the transforms above. We will limit our discussion to DFT, that is widely available as part of software packages like Matlab, Scipy(python) etc.., however we can approximate other transforms using DFT.

Real and Complex versions of transforms

For each of the listed transforms above, there exist a real version and complex version. The real version of the transform, takes in a real numbers and gives two sets of real frequency domain points – one set representing coefficients over $cosine$ basis function and the other set representing the co-efficient over $sine$ basis function. The complex version of the transforms represent positive and negative frequencies in a single array. The complex versions are flexible that it can process both complex valued signals and real valued signals. The following figure captures the difference between real DFT and complex DFT.

Real DFT:

Consider the case of N-point $real$ DFT , it takes in N samples of (real-valued) time domain waveform $x[n]$ and gives two arrays of length $N/2+1$ each set projected on cosine and sine functions respectively.

\[\begin{align} X_{re}[k] &= \phantom{-}\frac{2}{N} \displaystyle{ \sum_{n=0}^{N-1} x[n] \cdot cos\left( \frac{2 \pi k n}{N} \right)} \\ X_{im}[k] &= -\frac{2}{N} \sum_{n=0}^{N-1} \displaystyle{ x[n] \cdot sin\left( \frac{2 \pi k n}{N} \right)} \end{align}\]

Here, the time domain index $n$ runs from $0 \rightarrow N$, the frequency domain index $k$ runs from $0 \rightarrow N/2$

The real-valued time domain signal $x [n]$ can be synthesized from the real DFT pairs as

\[x[n] = \displaystyle{ \sum_{k=0}^{N/2} X_{re}[K] \cdot cos\left( \frac{2 \pi k n}{N} \right) – X_{im}[K] \cdot sin\left( \frac{2 \pi k n}{N} \right)}\]

Caveat: When using the synthesis equation, the values $X_{re}[0]$ and $X_{re}[N/2] $ must be divided by two. This problem is due to the fact that we restrict the analysis to real-values only. These type of problems can be avoided by using complex version of DFT.

Complex DFT:

Consider the case of N-point complex DFT, it takes in N samples of complex-valued time domain waveform $x[n]$ and produces an array $X[k]$ of length N.

\[ X[k]= \displaystyle{\sum_{n=0}^{N-1} x[n] e^{-j2 \pi k n/N}}\]

The arrays values are interpreted as follows

● $X[0]$ represents DC frequency component
● Next $N/2$ terms are positive frequency components with $X[N/2]$ being the Nyquist frequency (which is equal to half of sampling frequency)
● Next $N/2-1$ terms are negative frequency components (note: negative frequency components are the phasors rotating in opposite direction, they can be optionally omitted depending on the application)

The corresponding synthesis equation (reconstruct $x[n]$ from frequency domain samples $X[k]$) is

\[x[n]= \displaystyle{ \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j2 \pi k n/N}} \]

From these equations we can see that the real DFT is computed by projecting the signal on cosine and sine basis functions. However, the complex DFT projects the input signal on exponential basis functions (Euler’s formula connects these two concepts).

When the input signal in the time domain is real valued, the complex DFT zero-fills the imaginary part during computation (That’s its flexibility and avoids the caveat needed for real DFT). The following figure shows how to interpret the raw FFT results in Matlab that computes complex DFT. The specifics will be discussed next with an example.

Figure 2: Interpretation of frequencies in complex DFT output

Fast Fourier Transform (FFT)

The FFT function in Matlab is an algorithm published in 1965 by J.W.Cooley and J.W.Tuckey for efficiently calculating the DFT. It exploits the special structure of DFT when the signal length is a power of 2, when this happens, the computation complexity is significantly reduced. FFT length is generally considered as power of 2 – this is called $radix-2$ FFT which exploits the twiddle factors. The FFT length can be odd as used in this particular FFT implementation – Prime-factor FFT algorithm where the FFT length factors into two co-primes.

FFT is widely available in software packages like Matlab, Scipy etc.., FFT in Matlab/Scipy implements the complex version of DFT. Matlab’s FFT implementation computes the complex DFT that is very similar to above equations except for the scaling factor. For comparison, the Matlab’s FFT implementation computes the complex DFT and its inverse as

\[ \begin{align} X[k] &= \displaystyle{ \phantom {\frac{1}{N}}\sum_{n=0}^{N-1} x[n] e^{-j2 \pi k n/N}} \\ x[n] &= \displaystyle{ \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j2 \pi k n/N}} \end{align}\]

The Matlab commands that implement the above equations are FFT and IFFT) respectively. The corresponding syntax is as follows

X = fft(x,N) %compute X[k]
x = ifft(X,N) %compute x[n]

Interpreting the FFT results

Lets assume that the $x[n]$ is the time domain cosine signal of frequency $f_c=10Hz$ that is sampled at a frequency $f_s=32*fc$ for representing it in the computer memory.

fc=10;%frequency of the carrier
fs=32*fc;%sampling frequency with oversampling factor=32
t=0:1/fs:2-1/fs;%2 seconds duration
x=cos(2*pi*fc*t);%time domain signal (real number)
subplot(3,1,1); plot(t,x);hold on; %plot the signal
title('x[n]=cos(2 \pi 10 t)'); xlabel('n'); ylabel('x[n]');

Figure 3: A 2 seconds record of 10 Hz cosine wave

Lets consider taking a $N=256$ point FFT, which is the $8^{th}$ power of $2$.

N=256; %FFT size
X = fft(x,N);%N-point complex DFT
%output contains DC at index 1, Nyquist frequency at N/2+1 th index
%positive frequencies from index 2 to N/2
%negative frequencies from index N/2+1 to N

Note: The FFT length should be sufficient to cover the entire length of the input signal. If $N$ is less than the length of the input signal, the input signal will be truncated when computing the FFT. In our case, the cosine wave is of 2 seconds duration and it will have 640 points (a $10Hz$ frequency wave sampled at 32 times oversampling factor will have $2 \times 32 \times 10 = 640$ samples in 2 seconds of the record). Since our input signal is periodic, we can safely use $N=256$ point FFT, anyways the FFT will extend the signal when computing the FFT (see additional topic on spectral leakage that explains this extension concept).

Due to Matlab’s index starting at 1, the DC component of the FFT decomposition is present at index 1.

>>X(1)
 1.1762e-14   (approximately equal to zero)

That’s pretty easy. Note that the index for the raw FFT are integers from $1 \rightarrow N$. We need to process it to convert these integers to $frequencies$. That is where the $sampling$ frequency counts.

Each point/bin in the FFT output array is spaced by the frequency resolution $\Delta f$ that is calculated as

\[ \Delta f = \frac{f_s}{N} \]

where, $f_s$ is the sampling frequency and $N$ is the FFT size that is considered. Thus, for our example, each point in the array is spaced by the frequency resolution

\[\Delta f = \frac{f_s}{N} = \frac{32*f_c}{256} = \frac{320}{256} = 1.25 Hz \]

Now, the $10 Hz$ cosine signal will leave a spike at the 8th sample ($10/1.25=8$), which is located at index 9 (See next figure).

>> abs(X(8:10)) %display samples 7 to 9
ans = 0.0000  128.0000    0.0000

Therefore, from the frequency resolution, the entire frequency axis can be computed as

%calculate frequency bins with FFT
df=fs/N %frequency resolution
sampleIndex = 0:N-1; %raw index for FFT plot
f=sampleIndex*df; %x-axis index converted to frequencies

Now we can plot the absolute value of the FFT against frequencies as

subplot(3,1,2); stem(sampleIndex,abs(X)); %sample values on x-axis
title('X[k]'); xlabel('k'); ylabel('|X(k)|');
subplot(3,1,3); plot(f,abs(X)); %x-axis represent frequencies
title('X[k]'); xlabel('frequencies (f)'); ylabel('|X(k)|');

The following plot shows the frequency axis and the sample index as it is for the complex FFT output.

Figure 4: Magnitude response from FFT output plotted against sample index (top) and computed frequencies (bottom)

After the frequency axis is properly transformed with respect to the sampling frequency, we note that the cosine signal has registered a spike at $10 Hz$. In addition to that, it has also registered a spike at $256-8=248^{th}$ sample that belongs to negative frequency portion. Since we know the nature of the signal, we can optionally ignore the negative frequencies. The sample at the Nyquist frequency ($f_s/2$) mark the boundary between the positive and negative frequencies.

>> nyquistIndex=N/2+1;
>> X(nyquistIndex-2:nyquistIndex+2).'
ans =
1.0e-13 *
  -0.2428 + 0.0404i
  -0.1897 + 0.0999i
  -0.3784          
  -0.1897 - 0.0999i
  -0.2428 - 0.0404i

Note that the complex numbers surrounding the Nyquist index are complex conjugates and they represent positive and negative frequencies respectively.

FFTShift

From the plot we see that the frequency axis starts with DC, followed by positive frequency terms which is in turn followed by the negative frequency terms. To introduce proper order in the x-axis, one can use FFTshift function Matlab, which arranges the frequencies in order: negative frequencies $\rightarrow$ DC $\rightarrow$ positive frequencies. The fftshift function need to be carefully used when $N$ is odd.

For even N, the original order returned by FFT is as follows (note: all indices below corresponds to Matlab’s index)

● $X[1]$ represents DC frequency component
● $X[2]$ to $X[N/2]$ terms are positive frequency components
● $X[N/2+1]$ is the Nyquist frequency ($F_s/2$) that is common to both positive and negative frequencies. We will consider it as part of negative frequencies to have the same equivalence with the fftshift function.
● $X[N/2+1]$ to $X[N]$ terms are considered as negative frequency components

FFTshift shifts the DC component to the center of the spectrum. It is important to remember that the Nyquist frequency at the (N/2+1)th Matlab index is common to both positive and negative frequency sides. FFTshift command puts the Nyquist frequency in the negative frequency side. This is captured in the following illustration.

Figure 5: Role of FFTShift in ordering the frequencies

Therefore, when $N$ is even, ordered frequency axis is set as

\[f = \Delta f \left[ -\frac{N}{2}:1:\frac{N}{2}-1 \right] = \frac{f_s}{N} \left[ -\frac{N}{2}:1:\frac{N}{2}-1 \right]\]

When (N) is odd, the ordered frequency axis should be set as

\[ f = \Delta f \left[ -\frac{N-1}{2}:1:\frac{N+1}{2}-1 \right] = \frac{f_s}{N} \left[ -\frac{N-1}{2}:1:\frac{N-1}{2}-1 \right]\]

The following code snippet, computes the fftshift using both the manual method and using the Matlab’s in-build command. The results are plotted by superimposing them on each other. The plot shows that both the manual method and fftshift method are in good agreement.

%two-sided FFT with negative frequencies on left and positive frequencies on right
%following works only if N is even, for odd N see equation above
X1 = [(X(N/2+1:N)) X(1:N/2)]; %order frequencies without using fftShift
X2 = fftshift(X);%order frequencies by using fftshift

df=fs/N; %frequency resolution
sampleIndex = -N/2:N/2-1; %raw index for FFT plot
f=sampleIndex*df; %x-axis index converted to frequencies
%plot ordered spectrum using the two methods
figure;subplot(2,1,1);stem(sampleIndex,abs(X1));hold on; stem(sampleIndex,abs(X2),'r') %sample index on x-axis
title('Frequency Domain'); xlabel('k'); ylabel('|X(k)|');%x-axis represent sample index
subplot(2,1,2);stem(f,abs(X1)); stem(f,abs(X2),'r') %x-axis represent frequencies
xlabel('frequencies (f)'); ylabel('|X(k)|');

Figure 6: Magnitude response of FFT result after applying FFTShift : plotted against sample index (top) and against computed frequencies (bottom)

Comparing the bottom figures in the Figure 4 and Figure 6, we see that the ordered frequency axis is more meaningful to interpret.

IFFTShift

One can undo the effect of fftshift by employing ifftshift function. The ifftshift function restores the raw frequency order. If the FFT output is ordered using fftshift function, then one must restore the frequency components back to original order BEFORE taking IFFT. Following statements are equivalent.

X = fft(x,N) %compute X[k]
x = ifft(X,N) %compute x[n]

X = fftshift(fft(x,N)); %take FFT and rearrange frequency order (this is mainly done for interpretation)
x = ifft(ifftshift(X),N)% restore raw frequency order and then take IFFT

Some observations on FFTShift and IFFTShift

When $N$ is odd and for an arbitrary sequence, the fftshift and ifftshift functions will produce different results. However, when they are used in tandem, it restores the original sequence.

>>  x=[0,1,2,3,4,5,6,7,8]
0     1     2     3     4     5     6     7     8
>>  fftshift(x)
5     6     7     8     0     1     2     3     4
>>  ifftshift(x)
4     5     6     7     8     0     1     2     3
>>  ifftshift(fftshift(x))
0     1     2     3     4     5     6     7     8
>>  fftshift(ifftshift(x))
0     1     2     3     4     5     6     7     8

When $N$ is even and for an arbitrary sequence, the fftshift and ifftshift functions will produce the same result. When they are used in tandem, it restores the original sequence.

>>  x=[0,1,2,3,4,5,6,7]
0     1     2     3     4     5     6     7
>>  fftshift(x)
4     5     6     7     0     1     2     3
>>  ifftshift(x)
4     5     6     7     0     1     2     3
>>  ifftshift(fftshift(x))
0     1     2     3     4     5     6     7
>>  fftshift(ifftshift(x))
0     1     2     3     4     5     6     7

For Python code, please check the following book: Digital Modulations using Python – by Mathuranathan Viswanathan

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Topics in this chapter

Books by the author

Wireless Communication Systems in Matlab Second Edition(PDF) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Python (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart	Digital Modulations using Matlab (PDF ebook) Note: There is a rating embedded within this post, please visit this post to rate it. Checkout Added to cart
Hand-picked Best books on Communication Engineering Best books on Signal Processing

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