Fast Fourier Transform - Algorithms and Applications (Signals and Communication Technology)

High Performance Sparse Fast Fourier Transform

---

Fast Fourier Transform - Algorithms and Applications is designed for senior undergraduate and graduate students, faculty, engineers, and scientists in the field, and self-learners to understand FFTs and directly apply them to their fields, efficiently. It is designed to be both a text and a reference. By including many figures, tables, bock diagrams and graphs, this book helps the reader understand the concepts of fast algorithms readily and intuitively.

This book is for any professional who wants to have a basic understanding of the latest developments in and applications of FFT. It provides a good reference for any engineer planning to work in this field, either in basic implementation or in research and development. Notes Description based upon print version of record. Includes bibliographical references and index. Contents Fast Fourier Transform: Discrete Fourier Transform; 2. Integer Fast Fourier Transform; 4.

Two-Dimensional Discrete Fourier Transform; 5. You are browsing titles by their Library of Congress call number classification. Library Staff Details Staff view. Access Note UW-Madison does not have a subscription for electronic access to this title, which may only be available online to patrons at other UW System campuses. Ask a Librarian to determine your options for obtaining access. This section describes the problem of spectral leakage, the characteristics of windows, some strategies for choosing windows, and the importance of scaling windows. Another important characteristic of window spectra is main lobe width.

The frequency resolution of the windowed signal is limited by the width of the main lobe of the window spectrum. Therefore, the ability to distinguish two closely spaced frequency components increases as the main lobe of the window narrows. As the main lobe narrows and spectral resolution improves, the window energy spreads into its side lobes, and spectral leakage worsens. In general, then, there is a trade off between leakage suppression and spectral resolution. To simplify choosing a window, you need to define various characteristics so that you can make comparisons between windows.

Figure 11 shows the spectrum of a typical window. To characterize the main lobe shape, the -3 dB and -6 dB main lobe width are defined to be the width of the main lobe in FFT bins or frequency lines where the window response becomes 0. To characterize the side lobes of the window, the maximum side lobe level and side lobe roll-off rate are defined. The maximum side lobe level is the level in decibels relative to the main lobe peak gain, of the maximum side lobe.

The side lobe roll-off rate is the asymptotic decay rate, in decibels per decade of frequency, of the peaks of the side lobes. Table 1 lists the characteristics of several window functions and their effects on spectral leakage and resolution. Notice that this method is valid only for a spectrum made up of discrete frequency components.

It is not valid for a continuous spectrum. Also, if two or more frequency peaks are within six lines of each other, they contribute to inflating the estimated powers and skewing the actual frequencies. You can reduce this effect by decreasing the number of lines spanned by the preceding computations.

If two peaks are that close, they are probably already interfering with one another because of spectral leakage. Similarly, if you want the total power in a given frequency range, sum the power in each bin included in the frequency range and divide by the noise power bandwidth of the windows. Refer to the Computations on the Spectrum topic in the LabVIEW Help linked below for the most updated information about estimating power and frequency. The measurement of noise levels depends on the bandwidth of the measurement.

When looking at the noise floor of a power spectrum, you are looking at the narrowband noise level in each FFT bin. Thus, the noise floor of a given power spectrum depends on the f of the spectrum, which is in turn controlled by the sampling rate and number of points.

In other words, the noise level at each frequency line reads as if it were measured through a f Hz filter centered at that frequency line. Therefore, for a given sampling rate, doubling the number of points acquired reduces the noise power that appears in each bin by 3 dB. Discrete frequency components theoretically have zero bandwidth and therefore do not scale with the number of points or frequency range of the FFT.

To compute the SNR, compare the peak power in the frequencies of interest to the broadband noise level. Compute the broadband noise level in Vrms 2 by summing all the power spectrum bins, excluding any peaks and the DC component, and dividing the sum by the equivalent noise bandwidth of the window. For example, in Figure 6 the noise floor appears to be more than dB below full scale, even though the PCI Family dynamic range is only 93 dB. If you were to sum all the bins, excluding DC, and any harmonic or other peak components and divide by the noise power bandwidth of the window you used, the noise power level compared to full scale would be around dB from full scale.

Content Types

Because of noise-level scaling with f, spectra for noise measurement are often displayed in a normalized format called power or amplitude spectral density. This normalizes the power or amplitude spectrum to the spectrum that would be measured by a 1 Hz-wide square filter, a convention for noise-level measurements. The level at each frequency line then reads as if it were measured through a 1 Hz filter centered at that frequency line.

Power spectral density is computed as. The spectral density format is appropriate for random or noise signals but inappropriate for discrete frequency components because the latter theoretically have zero bandwidth. Refer to the Computations on the Spectrum topic in the LabVIEW Help linked below for the most updated information about computing noise levels and the power spectral density.

The cross power spectrum is in two-sided complex form. To convert to magnitude and phase, use the Rectangular-To-Polar conversion function. To convert to a single-sided form, use the same method described in the Converting from a Two-Sided Power Spectrum to a Single-Sided Power Spectrum section of this application note. The units of the single-sided form are in volts or other quantity rms squared. The power spectrum is equivalent to the cross power spectrum when signals A and B are the same signal.

Therefore, the power spectrum is often referred to as the auto power spectrum or the auto spectrum. The single-sided cross power spectrum yields the product of the rms amplitudes of the two signals, A and B, and the phase difference between the two signals. When you know how to use these basic blocks, you can compute other useful functions, such as the Frequency Response function.

Three useful functions for characterizing the frequency response of a network are the frequency response, impulse response, and coherence functions. The frequency response of a network is measured by applying a stimulus to the network as shown in Figure 12 and computing the frequency response from the stimulus and response signals. The frequency response function is in two-sided complex form. To convert to the frequency response gain magnitude and the frequency response phase, use the Rectangular-To-Polar conversion function.

To convert to single-sided form, simply discard the second half of the array. You may want to take several frequency response function readings and then average them. To do so, average the cross power spectrum, S AB f , by summing it in the complex form then dividing by the number of averages, before converting it to magnitude and phase, and so forth.

The power spectrum, S AA f , is already in real form and is averaged normally. The impulse response function of a network is the time-domain representation of the frequency response function of the network. To compute the impulse response of the network, take the inverse FFT of the two-sided complex frequency response function as described in the Frequency Response Function section of this application note.

The coherence function is often used in conjunction with the frequency response function as an indication of the quality of the frequency response function measurement and indicates how much of the response energy is correlated to the stimulus energy.

Fast Fourier Transform Algorithms and Applications

If there is another signal present in the response, either from excessive noise or from another signal, the quality of the network response measurement is poor. You can use the coherence function to identify both excessive noise and causality, that is, identify which of the multiple signal sources are contributing to the response signal. The coherence function is computed as. The result is a value between zero and one versus frequency. A zero for a given frequency line indicates no correlation between the response and the stimulus signal.

A one for a given frequency line indicates that the response energy is percent due to the stimulus signal; in other words, there is no interference at that frequency. For a valid result, the coherence function requires an average of two or more readings of the stimulus and response signals. For only one reading, it registers unity at all frequencies. To average the cross power spectrum, S AB f , average it in the complex form then convert to magnitude and phase as described in the Frequency Response Function section of this application note.

To achieve a good frequency response measurement, significant stimulus energy must be present in the frequency range of interest. Two common signals used are the chirp signal and a broadband noise signal. The chirp signal is a sinusoid swept from a start frequency to a stop frequency, thus generating energy across a given frequency range. White and pseudorandom noise have flat broadband frequency spectra; that is, energy is present at all frequencies. It is best not to use windows when analyzing frequency response signals. If you generate a chirp stimulus signal at the same rate you acquire the response, you can match the acquisition frame size to match the length of the chirp.

No window is generally the best choice for a broadband signal source. Because some stimulus signals are not constant in frequency across the time record, applying a window may obscure important portions of the transient response. There are many issues to consider when analyzing and measuring signals from plug-in DAQ devices. Unfortunately, it is easy to make incorrect spectral measurements. Understanding the basic computations involved in FFT-based measurement, knowing how to prevent antialiasing, properly scaling and converting to different units, choosing and using windows correctly, and learning how to use FFT-based functions for network measurement are all critical to the success of analysis and measurement tasks.

Being equipped with this knowledge and using the tools discussed in this application note can bring you more success with your individual application. Frequency Response and Network Analysis.

Rate this document Select a Rating 1 - Poor 2 3 4 5 - Excellent. This site uses cookies to offer you a better browsing experience. Learn more about our privacy policy. The content in this document might not reflect the most updated information available. For example, you can effectively acquire time-domain signals, measure the frequency content, and convert the results to real-world units and displays as shown on traditional benchtop spectrum and network analyzers. By using plug-in DAQ devices, you can build a lower cost measurement system and avoid the communication overhead of working with a stand-alone instrument.

Plus, you have the flexibility of configuring your measurement processing to meet your needs. To perform FFT-based measurement, however, you must understand the fundamental issues and computations involved. This application note serves the following purposes. Describes some of the basic signal analysis computations Discusses antialiasing and acquisition front ends for FFT-based signal analysis Explains how to use windows correctly Explains some computations performed on the spectrum Shows you how to use FFT-based functions for network measurement The basic functions for FFT-based signal analysis are the FFT, the Power Spectrum, and the Cross Power Spectrum.

Using these functions as building blocks, you can create additional measurement functions such as frequency response, impulse response, coherence, amplitude spectrum, and phase spectrum. FFTs and the Power Spectrum are useful for measuring the frequency content of stationary or transient signals. FFTs produce 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.

To measure frequency information that is changing over time, use joint time-frequency functions such as the Gabor Spectrogram. This application note also describes other issues critical to FFT-based measurement, such as the characteristics of the signal acquisition front end, the necessity of using windows, the effect of using windows on the measurement, and measuring noise versus discrete frequency components. Basic Signal Analysis Computations The basic computations for analyzing signals include converting from a two-sided power spectrum to a single-sided power spectrum, adjusting frequency resolution and graphing the spectrum, using the FFT, and converting power and amplitude into logarithmic units.

Two-Sided Power Spectrum of Signal. Thus is the length of the time record that contains the acquired time-domain signal. The signal in Figures 1 and 2 contains 1, points sampled at 1. The computations for the frequency axis demonstrate that the sampling frequency determines the frequency range or bandwidth of the spectrum and that for a given sampling frequency, the number of points acquired in the time-domain signal record determine the resolution frequency.

To increase the frequency resolution for a given frequency range, increase the number of points acquired at the same sampling frequency. For example, acquiring 2, points at 1. Alternatively, if the sampling rate had been Adequate and Inadequate Signal Sampling. For an accurate spectral measurement, it is not sufficient to use proper signal acquisition techniques to have a nicely scaled, single-sided spectrum. You might encounter spectral leakage. Spectral leakage is the result of an assumption in the FFT algorithm that the time record is exactly repeated throughout all time and that signals contained in a time record are thus periodic at intervals that correspond to the length of the time record.

If the time record has a nonintegral number of cycles, this assumption is violated and spectral leakage occurs. Another way of looking at this case is that the nonintegral cycle frequency component of the signal does not correspond exactly to one of the spectrum frequency lines. There are only two cases in which you can guarantee that an integral number of cycles are always acquired. One case is if you are sampling synchronously with respect to the signal you measure and can therefore deliberately take an integral number of cycles. Another case is if you capture a transient signal that fits entirely into the time record.

EXERCÍCIOS RESOLVIDOS

In most cases, however, you measure an unknown signal that is stationary; that is, the signal is present before, during, and after the acquisition. In this case, you cannot guarantee that you are sampling an integral number of cycles. Spectral leakage distorts the measurement in such a way that energy from a given frequency component is spread over adjacent frequency lines or bins. You can use windows to minimize the effects of performing an FFT over a nonintegral number of cycles.

Figure 7 shows the effects of three different windows -- none Uniform , Hanning also commonly known as Hann , and Flat Top -- when an integral number of cycles have been acquired, in this figure, cycles in a 1,point record. Notice that the windows have a main lobe around the frequency of interest. This main lobe is a frequency domain characteristic of windows. The Uniform window has the narrowest lobe, and the Hann and Flat Top windows introduce some spreading.

The Flat Top window has a broader main lobe than the others. For an integral number of cycles, all windows yield the same peak amplitude reading and have excellent amplitude accuracy. Figure 7 also shows the values at frequency lines of Hz through Hz for each window. The amplitude error at Hz is 0 dB for each window. The graph shows the spectrum values between and Hz.

The actual values in the resulting spectrum array for each window at through Hz are shown below the graph. Figure 8 shows the leakage effects when you acquire Notice that at a nonintegral number of cycles, the Hann and Flat Top windows introduce much less spectral leakage than the Uniform window. Also, the amplitude error is better with the Hann and Flat Top windows. The Flat Top window demonstrates very good amplitude accuracy but also has a wider spread and higher side lobes than the Hann window. Power Spectrum of 1 Vrms Signal at In addition to causing amplitude accuracy errors, spectral leakage can obscure adjacent frequency peaks.

Figure 9 shows the spectrum for two close frequency components when no window is used and when a Hann window is used. To understand how a given window affects the frequency spectrum, you need to understand more about the frequency characteristics of windows. The windowing of the input data is equivalent to convolving the spectrum of the original signal with the spectrum of the window as shown in Figure Even if you use no window, the signal is convolved with a rectangular-shaped window of uniform height, by the nature of taking a snapshot in time of the input signal.

This convolution has a sine function characteristic spectrum. For this reason, no window is often called the Uniform or Rectangular window because there is still a windowing effect. An actual plot of a window shows that the frequency characteristic of a window is a continuous spectrum with a main lobe and several side lobes. The main lobe is centered at each frequency component of the time-domain signal, and the side lobes approach zero at. Frequency Characteristics of a Windowed Spectrum.

Read this book

An FFT produces a discrete frequency spectrum. The continuous, periodic frequency spectrum is sampled by the FFT, just as the time-domain signal was sampled by the ADC. What appears in each frequency line of the FFT is the value of the continuous convolved spectrum at each FFT frequency line. This is sometimes referred to as the picket-fence effect because the FFT result is analogous to viewing the continuous windowed spectrum through a picket fence with slits at intervals that corresponds to the frequency lines. If the frequency components of the original signal match a frequency line exactly, as is the case when you acquire an integral number of cycles, you see only the main lobe of the spectrum.

Side lobes do not appear because the spectrum of the window approaches zero at f intervals on either side of the main lobe. Figure 7 illustrates this case. If a time record does not contain an integral number of cycles, the continuous spectrum of the window is shifted from the main lobe center at a fraction of f that corresponds to the difference between the frequency component and the FFT line frequencies. This shift causes the side lobes to appear in the spectrum. In addition, there is some amplitude error at the frequency peak, as shown in Figure 8, because the main lobe is sampled off center the spectrum is smeared.

Figure 11 shows the frequency spectrum characteristics of a window in more detail. The side lobe characteristics of the window directly affect the extent to which adjacent frequency components bias leak into adjacent frequency bins. The side lobe response of a strong sinusoidal signal can overpower the main lobe response of a nearby weak sinusoidal signal. Frequency Response of a Window. Each window has its own characteristics, and different windows are used for different applications. To choose a spectral window, you must guess the signal frequency content. If the signal contains strong interfering frequency components distant from the frequency of interest, choose a window with a high side lobe roll-off rate.

If there are strong interfering signals near the frequency of interest, choose a window with a low maximum side lobe level. If the frequency of interest contains two or more signals very near to each other, spectral resolution is important. In this case, it is best to choose a window with a very narrow main lobe. If the amplitude accuracy of a single frequency component is more important than the exact location of the component in a given frequency bin, choose a window with a wide main lobe.

• The Multinational Enterprise (RLE International Business) (Routledge Library Editions: International Business)
• Dragonflies Chasing Mosquitoes
• Toronto: The Way We Were
• 30 Days of Conviction: A No Frills Straight Truth Guide to True Salvation
• JOY - All The Bible Teaches About