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Fourier Transform – FFT – IFFT
Using the Fourier Transform Calculator
The Fourier Transform Calculator performs FFT and IFFT transforms of input data, and displays time-domain and frequency-domain representations of that data in graphical form. The Fourier Transform Calculator accepts two different sources of data: Preset Data Objects The menu entitled "Time-Domain Input Function" provides many preset time-domain data objects that can be processed by the Fourier Transform Calculator. To use this source of data, simply select one of the data objects from that menu. The Fourier Transform Calculator will automatically process that data object via the FFT, and display the frequency-domain spectrum for that data object. The menu entitled "FFT Number of Samples" applies only to the Preset Data Objects. Use that menu to choose the number of samples that are taken from the Preset Data Object currently selected in the "Time-Domain Input Function" menu. Data Samples Enter a list of data samples in the first input box, and the sampling rate of that data in the second input box, then click the Calculate button. The Fourier Transform Calculator will automatically process your input data via the FFT, and display the frequency-domain spectrum for that input data. The number of samples is automatically obtained from the list of data samples you provide. For best results in the FFT, you should provide a number of data samples that is a power-of-2, such as 256, 512, etc. However, if the length of your data list is not a power-of-2, the calculator will automatically zero-pad your input data to the next higher power-of-2.List of Data Samples
To Fourier transform your data samples, enter a list of samples, with each sample separated from the next by a blank-space or a comma. Data Samples List Format The format of your samples list must be one of these: s1, s2, ... sN-1, sN or s1 s2 ... sN-1 sN where N is the total number of data samples. Fast Fourier Transform Your data samples are processed by a discrete digital version of the continuous Fourier Transform. The discrete or digital version is called a Fast Fourier Transform, or FFT. Size of Data Samples List For best results, the number of samples in your input list should be a power of 2, such as 1024, 4096, 32768, upto a maximum of: 2^18 = 256 * 1024 = 262144 Each number in your data sample list should be in one of the following formats. You can mix the formats: Integer: 31457 Decimal: +9862.314 Scientific Notation: 2.7E+2 Your list should look like one of these: 2, 2.7E+2, 2.718, 0.35e03, ... 2 2.7E+2 2.718 0.35e03 ...FFT Frequency Domain Output
FFT Frequency Domain Spectral Graph The result of FFT processing is a frequency domain spectrum, diplayed as a graph. The peaks that may appear in the frequency domain graph indicate the presence of important frequency components in the original analog signal. X-Axis of the Spectral Graph The x-axis represents frequency in cycles per second, or Hertz. Y-Axis of the Spectral Graph The y-axis represents magnitude of the frequency components in decibels. Decibels (dB) are a logarithmic measure of, in this case, the amplitude of a "field" value. "Field" value decibels traditionally represent the following calculation: dB = 20 * log10( magnitude ) This calculation is proportional to the square of the amplitude of the "field" value, and typically also proportional to Power, when the "field" value represents a physical quantity such as Voltage.FFT Frequency Domain Spectral Graph
Optional Y-Axis Magnitude The y-axis of the frequency domain graph is set to represent magnitude in decibels, as discussed above. However, you can change that setting via this menu: Select Y-Axis Magnitude Menu This menu has the following options: Sqrt( R^2 + I^2 ) Square root of: the sum of the squared Real (R) component of the FFT result and the squared Imaginary (I) component of the FFT result. Real and Real^2 The Real component of the FFT result, and the square of the Real component. Imaginary and Imaginary^2 The Imaginary component of the FFT result, and the square of the Imaginary component. dB of Sqrt( R^2 + I^2 ) The decibel of the square root of: the sum of the squared Real component of the FFT result and the squared Imaginary component of the FFT result. dB Real^2 and dB Imag^2 The decibels of the squared Real component of the FFT result, and the decibels of the squared Imaginary component of the FFT result. Normalized Decibel Values Similar to the above decibel values, but normalized so that the peak value is at 0 dB.Input Data and Window Functions
FFT Window Functions1 The FFT can pre-multiply the input time-domain data with any of several window functions. Window functions are commonly used in signal processing to reduce the "spectral leakeage" caused when the FFT is applied to a typically non-periodic finite-length sequence of input data. To apply a window function, simply select one of the window functions from the "FFT Window Function" menu. The selected window function is automatically applied to the sample data prior to the FFT. Time-Domain Input Function Input data for the FFT can be supplied as a Sample List, via this input box, or by one of the preset Data Objects available in the menu entitled "Time-Domain Input Function". That menu contains an assortment of preset Data Objects, including all of the window functions. Note that window functions are provided in the "Time-Domain Input Function" menu, so that users can see, review and analyze the time-domain and frequency-domain characteristics of those window functions. However, to pre-multiply a window function with the input data, use the "FFT Window Function" menu, rather than the "Time-Domain Input Function" menu. The FFT Window Functions The following window functions are provided: Rectangle, Triangle, Hann (aka Hanning, Cosine^1), "Raised Cosine" 2, 3, and 4, Hamming, Riesz, Tukey, Riemann, de La Vallee Poussin (aka Parzen), Bohman, Poussin, Hanning-Poussin, Cauchy, Cosine, Gaussian, Kaiser-Bessel, Blackman Exact, Blackman, Blackman-Harris, Lanczos, Bartlett, Bartlett-Hann, Nuttall, Blackman-Nuttall, and Flat Top. References: [1] Harris, Fredric J. "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform." Proceedings of the IEEE. Vol. 66, No. 1 (January 1978).Sampling Rate of your Data
Sampling Rate Enter the sampling rate of your input data, in Hertz, (i.e. samples per second.) Sampling Rate Format The sampling rate value that you enter should be in one of the following formats: Integer: 8820 Decimal: 44100.9 Scientific Notation: 1.102536e04 Nyquist–Shannon Sampling Theorem According to the Nyquist–Shannon theorem, a band-limited analog signal can only be completely reconstructed (via the inverse Fourier transform), if it is sampled at a rate of "2f" or higher, where "f" is the highest frequency contained in the original signal. Therefore, you must sample your source analog signal at a sampling rate of at least twice the maximum frequency you want to capture. For example, to capture a 3 Kilohertz frequency component in your data, you must sample the analog signal at a sampling rate of at least 6 Kilohertz. Limitations of the Fast Fourier Transform For best results, the number of samples in your input data list should be a power of 2, such as 1024, 4096, 32768, upto to a maximum of: 2^18 = 256 * 1024 = 262144 Due to the internal workings of the FFT, data sample lists larger than 2^18 require significant amounts of computer processor and memory resources, and are disallowed as of this writing.Enter the Graph Title
If you want your graph to have a title, enter it here. This is optional. The graph title length is limited to 30 characters maximum.Enter the Graph x-Axis Label
If you want your graph to have an x-axis label, enter it here. This is optional. The graph x-axis label length is limited to 50 characters maximum.Enter the Graph y-Axis Label
If you want your graph to have a y-axis label, enter it here. This is optional. The graph y-axis label length is limited to 50 characters maximum.Control Buttons
Calculate Button – Use any of the following methods to perform the Fast Fourier Transform (FFT) of the data you entered, and automatically graph the frequency domain spectrum of your Fourier transformed data.- Click the Calculate button
- Press Enter or Return on keyboard
Control Buttons
Reflect X Button – This button reflects, or un-reflects, the x-axis of the frequency-domain spectrum, so that the negative frequencies are shown, or hidden. Zoom In Button – This button enables and disables "Zoom In" mode. When enabled, the mouse cursor changes to the "+" cursor, and you can click and drag on the graph area to zoom in. Zoom Out Button – This button enables and disables "Zoom Out" mode. When enabled, the mouse cursor changes to the "-" cursor, and you can click and drag on the graph area to zoom out. Time / Frequency Domain Button – This button switches the graph view between the time-domain and frequency-domain representations of the input data. FFT / Inv-FFT Button – This button runs the FFT (forward Fast Fourier Transform), or the IFFT (inverse Fast Fourier Transform), on the input data. Note that the FFT (forward Fast Fourier Transform), automatically runs whenever you select a new item from any of the menus.All Math Tools
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