Fast Fourier Transform Analysis

1. Fast Fourier Transform

2. Created by :
Farhan Wahyu Pratama 15/379992/TK/43257
Farras Muhammad Fikry 15/379993/TK/43258
Muhammad Rifaldi Aldino Fahreza 15/379998/TK/43263
Tito Setyadi Wiguna 15/380003/TK/43268

3. MENU
FFT
WHAT IS IT? TYPES SPECTRUM CHARACTERISTIC

4. WHAT IS IT...
• FFT is Fast Fourier Transform. The FFT is a faster version of the Discrete
• Fourier Transform (DFT). The FFT utilizes some clever algorithms to do the same
thing as the DTF, but in much less time.
• FFT is an efficient algorithm for calculating the Discrete Fourier Transform. It
calculates the exact same result (with possible minor differences due to rounding
of intermediate results)
• FFTs and the Power Spectrum are useful for measuring the frequency content of
stationary or transient signals. FFTsproduce the average frequency content of a
signal over the entire time that the signal was acquired. For this reason, you
should use FFTs for stationary signal analysis or in cases where you need only the
average energy at each frequency line.
Before we know about FFT we have to know about DFT.

5. DFT
The DFT (Discrete Fourier Transform) is extremely important in the area of
frequency (spectrum) analysis because it takes a discrete signal in the time domain
and transforms that signal into its discrete frequency domain representation.
Without a discrete-time to discrete-frequency transform. we would not be able to
compute the Fourier transform with a microprocessor or DSP based system. It is the
speed and discrete nature of the FFT that allows us to analyze a signal's spectrum
with Matlab or in real-time on the SR770
Source : University of Rhode Island Department of Electrical
and Computer Engineering
ELE 436: Communication Systems - FFT Tutorial
MENU

6. TYPES
• The form of FFT we have described is called “decimation in time”; there is a form called
“decimation in frequency” (but it has no advantages).
• The "radix 2" FFT must have length N a power of 2. Slightly more efficient is the "radix 4" FFT, in
which 2-input 2-output butterflies are replaced by 4-input 4-output units. The transform length
must then be a power of 4 (more restrictive).
• A completely different type of algorithm, the Winograd Fourier Transform Algorithm (WFTA), can
be used for FFT lengths equal to the product of a number of mutually prime factors (e.g. 9*7*5 =
315 or 5*16 = 80). The WFTA uses fewer multipliers, but more adders, than a similar-length FFT.
• Efficient algorithms exist for FFTing real (not complex) data at about 60% the effort of the same-
sized complex-data FFT.
Source : Fast Fourier Transform (FFT) by Elena Punskaya
MENU

7. SPECTRUM
The FFT does not directly give you the spectrum of a signal. The FFT can
vary dramatically depending on the number of points (N) of the FFT, and the
number of periods of the signal that are represented. There is another problem as
well. The FFT contains information between 0 and fs, however, we know that the
sampling frequency must be at least twice the highest frequency component.
Therefore, the signal's spectrum shouldbe entirly below
fs
2
, the Nyquist frequency.
Recall also that a real signal should have a transform magnitude that is
symmetrical for positive and negative frequencies. So instead of having a spectrum
that goes from 0 to fs, it would be more appropriate to show the spectrum from
−fs
2
to
fs
2
. This can be accomplished by using Matlab's fftshift.

8. Matlab's fftshift function as the following code demonstrates.
n = [0:149];
x1 = cos(2*pi*n/10);
N = 2048;
X = abs(fft(x1,N));
X = fftshift(X);
F = [-N/2:N/2-1]/N;
plot(F,X),
xlabel('frequency / f_s')
Source : University of Rhode Island Department of Electrical
and Computer Engineering
ELE 436: Communication Systems - FFT Tutorial
Approximate Spectrum of a Sinusoid with the FF

9. The Effect of Data Length can effect the resolution
Source : Fast Fourier Transform (FFT) by Elena Punskaya
MENU

10. CHARACTERISTIC
Advantage of FFT spectrum analyzer technology
• Fast capture of waveform: In view of the fact that the waveform is analysed digitally, the
waveform can be captured in a relatively short time, and then the subsequently analysed. This
short capture time can have many advantages - it can allow for the capture of transients or short
lived waveforms.
• Able to capture non-repetitive events: The short capture time means that the FFT analyzer can
capture non-repetitive waveforms, giving them a capability not possible with other spectrum
analyzers.
• Able to analyse signal phase: As part of the signal capture process, data is gained which can be
processed to reveal the phase of signals.
• Waveforms can be stored : Using FFT technology, it is possible to capture the waveform and
analyse it later should this be required.

11. MENU
Disadvantages of the FFT spectrum analyzer technology
• Frequency limitations: The main limit of the frequency and bandwidth of FFT
spectrum analyzers is the analogue to digital converter, ADC that is used to
convert the analogue signal into a digital format. While technology is improving
this component still places a major limitation on the upper frequency limits or
the bandwidth if a down-conversion stage is used.
• Cost: The high level of performance required by the ADC means that this item is
a very high cost item. In addition to all the other processing and display circuitry
required, this results in the costs rising for these items.
ENDSource: http://www.radio-electronics.com

12. TERIMA KASIH