No.41 FFT and frequency resolution
In FFT Analysis (Fourier analysis), the time axis waveform is cut out to a certain length, and the cut out time axis waveform is transformed to obtain the Fourier series. The Fourier series is arranged on the frequency axis, which is called the Fourier spectrum.
Time window length and frequency resolution
The relations with the length of time-axis waveform (Time window length [sec]), frequency resolution [Hz], frequency range, sampling points are as follows:
Number of line [point] = Sampling points [point] ÷ 2.56
Frequency resolution [Hz] = Frequency range [Hz] ÷ Number of line [point]
Time window length [sec] = 1 ÷ Frequency resolution [Hz]
If frequency range is 8 kHz, sampling points is 2048:
Number of line is 800 points
Time window length is 0.1 sec,
Frequency resolution is 10 Hz.
The time waveform with length of 0.1 seconds is used for an FFT analysis. The spectrum values are obtained by performing FFT every 10 Hz. The spectrum cannot be obtained with the fractional frequency component values, such as 995, 1001 Hz.
Spectrum of signals that deviate from the frequency resolution
When the sine wave with amplitude of 1 V and frequency of 1 kHz is performed by FFT under frequency range of 8 kHz, sampling points 2048 points, the spectrum value of 1 kHz component is 1V.
When FFT analysis is performed on signals that deviate from the frequency resolution, what spectrum you obtain depends on the window function. When the sine wave with amplitude of 1 V and frequency of 995 Hz is performed by FFT using Hanning window under frequency range of 8 kHz, sampling points 2048 points, the spectrum value of 990 Hz and 1 kHz components are approximately 0.85 V. The amplitude of the frequency signals that deviate from the frequency resolution (10 Hz) is divided into the frequency components on both sides.
Example of spectrum of signals that deviate from the frequency resolution
Table 1 shows the values of each frequency components from 980 Hz to 1020 Hz when the FFT analysis is performed by changing the frequency of the sine wave with amplitude, 1 from 990 Hz to 1010 Hz. The value of the sum of squares represents the sum of squares of these five components. In addition, the value of the square root of the sum of squares is calculated. The Hanning window is used as window function.
Table 1 FFT analysis result of Sine wave（990 to 1010 Hz）
| Frequency of Sine wave | 980 Hz | 990 Hz | 1000 Hz | 1010 Hz | 1020 Hz | Sum of squares | Square root of the sum of squares |
|---|---|---|---|---|---|---|---|
| 990 Hz | 0.500 | 1.000 | 0.500 | 0.000 | 0.000 | 1.500 | 1.225 |
| 991 Hz | 0.430 | 0.994 | 0.571 | 0.018 | 0.004 | 1.500 | 1.225 |
| 992 Hz | 0.352 | 0.974 | 0.652 | 0.047 | 0.010 | 1.500 | 1.225 |
| 993 Hz | 0.289 | 0.945 | 0.719 | 0.079 | 0.015 | 1.499 | 1.225 |
| 994 Hz | 0.222 | 0.898 | 0.792 | 0.124 | 0.021 | 1.499 | 1.224 |
| 995 Hz | 0.170 | 0.849 | 0.849 | 0.170 | 0.024 | 1.499 | 1.224 |
| 996 Hz | 0.124 | 0.792 | 0.898 | 0.222 | 0.026 | 1.500 | 1.225 |
| 997 Hz | 0.079 | 0.719 | 0.944 | 0.289 | 0.026 | 1.500 | 1.225 |
| 998 Hz | 0.047 | 0.652 | 0.974 | 0.289 | 0.022 | 1.500 | 1.225 |
| 999 Hz | 0.018 | 0.571 | 0.994 | 0.430 | 0.013 | 1.500 | 1.225 |
| 1000 Hz | 0.000 | 0.500 | 1.000 | 0.500 | 0.000 | 1.500 | 1.225 |
| 1001 Hz | 0.014 | 0.426 | 0.994 | 0.575 | 0.020 | 1.500 | 1.225 |
| 1002 Hz | 0.022 | 0.354 | 0.975 | 0.650 | 0.046 | 1.500 | 1.225 |
| 1003 Hz | 0.026 | 0.287 | 0.943 | 0.721 | 0.080 | 1.500 | 1.225 |
| 1004 Hz | 0.027 | 0.225 | 0.901 | 0.788 | 0.121 | 1.500 | 1.225 |
| 1005 Hz | 0.024 | 0.170 | 0.849 | 0.849 | 0.170 | 1.499 | 1.225 |
| 1006 Hz | 0.020 | 0.121 | 0.788 | 0.901 | 0.225 | 1.499 | 1.224 |
| 1007 Hz | 0.015 | 0.080 | 0.721 | 0.943 | 0.287 | 1.499 | 1.225 |
| 1008 Hz | 0.010 | 0.046 | 0.650 | 0.975 | 0.354 | 1.500 | 1.225 |
| 1009 Hz | 0.005 | 0.020 | 0.575 | 0.994 | 0.426 | 1.500 | 1.225 |
| 1010 Hz | 0.000 | 0.000 | 0.500 | 1.000 | 0.500 | 1.500 | 1.225 |
Looking at the 1000 Hz component of the sine wave at 1000 Hz, its amplitude is 1, which correctly indicates the amplitude of the sine wave. As for 990 Hz and 1010 Hz components, it is 0.5. This is because the spectrum is widened horizontally due to using the Hanning window. The sum of squares of 980 Hz to 1020 Hz is 1.5. This is same as above. This 1.5 is the value called Equivalent Noise Band Width, which indicates how much spectrum is widened.
Looking at 990 Hz and 1000 Hz component of 995 Hz sine wave, its amplitude is 0.849. This is because the magnitude of the sine wave is divided into the frequency components on both sides.
Even for a sine wave with the frequency that is not an integral multiple of the frequency resolution (10 Hz), the magnitude of the component is divided into the components on both sides and is smaller than 1. However, the sum of squares of the components next to both 5 Hz is 1.5. Divide this by the equivalent noise band width (1.5) and then take the square root to obtain the exact amplitude of the sine wave (1).
Summary
This time, I explained the result of FFT Analysis (Fourier analysis) of a sine wave using the Hanning window function. The spectrum is obtained for each frequency resolution in the FFT analysis. If the frequency of the sine wave is not an integral multiple of the frequency resolution, its amplitude will be divided into the frequency components on both sides and will be a smaller value. If you want to obtain an accurate value, it is necessary to change the setting that the frequency resolution is finer.
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