"Waveforms and FFT-10" Search Enhancement and Zoom Functions

8. Search Enhance and Zoom Functions

There are times when you want to read the frequency more accurately, such as when the input signal is a sine wave. One way to do this is:

• Even with the same frequency range, set a larger number of sample points.
• Search enhancement function (corrects frequency resolution for reading)
• Zoom function (Performs FFT with a specified center frequency and frequency bandwidth)

There is such a feature. This time, I will explain this function.

8-1 Sample Score

Figure 8-1 shows the power spectra obtained by performing an FFT on a 502Hz sine wave input signal with a frequency range of 1kHz and 256, 2048, and 16384 sample points. The frequency resolution is 1/100, 1/800, and 1/6400 of 1kHz, respectively, and with 16384 points, the X-axis value of the power spectrum is the maximum value = 502.031Hz, allowing for a more detailed frequency reading.

As explained previously, with the DS-0221 FFT analysis software:

Frequency resolution = Frequency range ÷ (Number of sample points ÷ 2.56)

It will be.

The upper part of Figure 8-1 has a sample size of 256 and a scale with coarse X-axis resolution, unlike the middle and lower parts.
It might seem that way, but the window function is the same Hanning window, so I explained it in the previous issue.
The shape of the bandpass filter remains unchanged.

8-2 Search Enhancement Function

The search enhancement function is only effective in the case of a Hanning window. The frequency of the search position is
We obtain the frequency resolution by multiplying it by 32 from the shape of the bandpass filter.
This function is called the "Search Enhancement Function," and it is our company's original technology.

The right-hand figure in Figure 8-2 shows the search enhancement function turned ON using the spectral data with 256 sample points from Figure 8-1. The left-hand figure shows the search enhancement function OFF. With 256 sample points, the frequency resolution is 10 Hz, so when reading the frequency using search, after 500 Hz is...
The frequency will be 510Hz, resulting in readings with 10Hz intervals.

With the search enhancement function ON, the frequency resolution becomes 1kHz/(100×32) = 0.3125Hz,
The frequency of the position is displayed as: #X: 501.875Hz #Y: -0.04dB, and the search enhancement function is also available.
Compared to not using it, you can read a value that is closer to the input signal of 502Hz.
When the search enhancement function is ON, it is displayed with the #X symbol.

8-3 Zoom function

One way to improve resolution regardless of the window function is through digital zoom.
In the DS-0221 FFT analysis software, this is simply referred to as the "zoom function."
The zoom function displays the spectrum of a specific frequency range f1 to f2 using a normal FFT, from 0 to f (frequency range).
The frequency is converted to be displayed within the wavenumber range (Hz). The width of Δ = f2 - f1 is changed to the width of f.
Because the image is magnified (Δ・M = f), it becomes possible to observe frequencies in more detail.

Figure 8-3 shows the results of performing an FFT on a 502Hz sine wave input signal with a frequency range of 1kHz and 256 sample points, zoomed to a width of 500Hz ± 10Hz, with the zoom function ON. The figure above shows the result with the zoom function OFF. The X-axis divides the range from 0 to 1kHz into 100 equal parts, with a frequency resolution of 10Hz for the power spectrum.
A circle is displayed. With the zoom function ON, the frequency band from 490Hz to 510Hz is also divided into 100 equal parts.
(Resolution 20Hz/100 = 0.2Hz) is displayed, and the frequency resolution is 50 times greater, and the input signal frequency
A frequency of 502Hz can be read more accurately.

The zoom function first passes the input signal through a narrowband bandpass filter with set frequencies f1 to f2. This signal is then multiplied by a cosine wave of frequency f1, shifting the frequency from -Δ to +Δ.
Since generated components are also produced, aliasing is used to remove them, just as in a normal FFT.
The signal is resampled through a filter at a frequency 2.56 times that of Δ. This series of processes is then performed.
It is performed using digital operations and then an FFT is performed on the resampled sample points N.

It's important to note that the frequency resolution is determined by Δ and the number of sample points N, and compared to the frequency range f, M = f/Δ times. If we consider the number of resample points N over time T (since T = N ÷ sample frequency), the time length will be M times longer than when the frequency range is f. Also, when the zoom function is ON, the time-axis waveform displayed is a frequency-shifted waveform, not the input waveform.

8-4 Changes in the spectrum when frequency resolution is increased

The width of a bandpass filter with a frequency resolution of 10 Hz is equivalent to 50 bandpass filters with a frequency resolution of 0.2 Hz.
This will be the width. Since we've been explaining using sine waves this time, the size of the spectrum is the same,
The frequencies of typical waveforms, especially random waveforms, are widely distributed, resulting in high frequency resolution.
Because the power spectrum is displayed in a more granular manner, the power spectrum appears smaller.
When viewing the power spectrum of a random signal, pay attention to the frequency resolution as well.
Figure 8-4 shows the power spectra of a random signal with 256 and 2048 sample points.

8-5 Power spectral density

For random signals, even if the frequency range is the same, the power spectrum value changes when the frequency resolution changes.
It does change. One way to ensure the same power spectrum value even when the frequency resolution changes is to use Power Spectrum Density (PSD). This is done by dividing the power spectrum obtained by FFT by the frequency resolution, as shown in the following formula, to normalize and display it as the power spectrum per 1 Hz. If you select a frequency range of 800 Hz and 2048 sample points, the resolution will be 1 Hz. However, using Power Spectrum Density is convenient because it displays the power spectrum per 1 Hz for any range. Note that the overall value will be the same as when using the power spectrum.

PSD: Spectral Density
Pi: Linear power spectrum (rms)
W: Window function correction value

• Rectangular window 1

• Hanning Window 3/2

• Flat top window 3.6714

Figure 8-5 shows the power spectrum of a random signal. The left column shows the power spectrum for 256 sample points, and the right column shows the power spectrum for 2048 sample points. The top row shows the power spectrum, and the bottom row shows the power spectral density (PSD). In the power spectral density display, there is no difference in spectral values depending on the number of sample points.

(Excerpt from the email newsletter issuedonSeptember20,2007)