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i_sm018_003
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Memory 1
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Source: http://www.sooeet.com © 2024 Sooeet.com All rights reserved
Online FFT Calculator
Graph Text for
User Data
Source
Title for the graph (optional):
Enter a Title for the Graph
If you want your graph to have a title, enter it here. This is optional.
If you leave this box empty, a suitable title is provided automatically.
The length of the title is limited to 80 characters maximum.
© 2024 Sooeet.com All rights reserved
Name for horizontal space-time axis (optional):
Name for Horizontal Space-Time Axis
If you want your graph to have a name for the horizontal space-time axis, enter it here. This is optional.
If you leave this box empty, a suitable name for the horizontal space-time axis is provided automatically.
The length of the name for the horizontal space-time axis is limited to 80 characters maximum.
© 2024 Sooeet.com All rights reserved
Name for horizontal frequency axis (optional):
Name for Horizontal Frequency Axis
If you want your graph to have a name for the horizontal frequency axis, enter it here. This is optional.
If you leave this box empty, a suitable name for the horizontal frequency axis is provided automatically.
The length of the name for the horizontal frequency axis is limited to 80 characters maximum.
© 2024 Sooeet.com All rights reserved
Name for vertical space-time axis (optional):
Name for Vertical Space-Time Axis
If you want your graph to have a name for the vertical space-time axis, enter it here. This is optional.
If you leave this box empty, a suitable name for the vertical space-time axis is provided automatically.
The length of the name for the vertical space-time axis is limited to 60 characters maximum.
© 2024 Sooeet.com All rights reserved
Name for vertical frequency axis (optional):
Name for Vertical Frequency Axis
If you want your graph to have a name for the vertical frequency axis, enter it here. This is optional.
If you leave this box empty, a suitable name for the vertical frequency axis is provided automatically.
The length of the name for the vertical frequency axis is limited to 60 characters maximum.
© 2024 Sooeet.com All rights reserved
Enter list of User-Data samples:
List of
User Data
Samples
To perform the Fast Fourier Transform on your data, enter a list of numbers, with each number in the list separated from the next number by a blank-space or a comma.
Format of Input Data
Your input data can be entered as a list of numbers, as in the following examples:
s1, s2, ... sN-1, sN
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.
Number of
User Data
Samples
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 = 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 ...
© 2024 Sooeet.com All rights reserved
Enter sampling rate:
Sampling Rate of Input Data
Sampling Rate
Enter the sampling rate of the input data, in Hertz, (i.e. samples per second.)
Sampling Rate Format
The sampling rate should be in one of the following formats:
Integer:
8820
Decimal:
44100.9
Scientific Notation:
1.1025e04
Nyquist–Shannon Sampling Theorem
The Nyquist–Shannon theorem is one of the principal elements of information theory. The theorem states that a bandlimited analog signal can be completely reconstructed via the inverse Fourier transform, only if the signal is sampled at a rate that exceeds
2F
, where
F
is the highest frequency contained in the original signal.
Therefore, when sampling a bandlimited analog signal intended for processing via the
FFT
, the sampling rate should exceed twice the maximum frequency that you wish to resolve with the
FFT
.
For example, to capture a 3 Kilohertz frequency component in the original analog signal, the analog signal must be sampled at a rate that exceeds 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 a maximum of:
2^18 = 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.
© 2024 Sooeet.com All rights reserved
User-Data Examples (optional):
Alto Saxophone D-flat 3
Alto Saxophone D3
Alto Saxophone E-flat 3
Alto Saxophone E3
Alto Saxophone F3
Alto Saxophone G-flat 3
Alto Saxophone G3
Alto Saxophone A-flat 3
Alto Saxophone A3
Alto Saxophone B-flat 3
Alto Saxophone B3
Alto Saxophone C4
Alto Saxophone D-flat 4
Alto Saxophone D4
Alto Saxophone E-flat 4
Alto Saxophone E4
Alto Saxophone F4
Alto Saxophone G-flat 4
Alto Saxophone G4
Alto Saxophone A-flat 4
Alto Saxophone A4
Alto Saxophone B-flat 4
Alto Saxophone B4
Alto Saxophone C5
Alto Saxophone D-flat 5
Alto Saxophone D5
Alto Saxophone E-flat 5
Alto Saxophone E5
Alto Saxophone F5
Alto Saxophone G-flat 5
Alto Saxophone G5
Alto Saxophone A-flat 5
Bass Clarinet D-flat 2
Bass Clarinet D2
Bass Clarinet E-flat 2
Bass Clarinet E2
Bass Clarinet F2
Bass Clarinet G-flat 2
Bass Clarinet G2
Bass Clarinet A-flat 2
Bass Clarinet A2
Bass Clarinet B-flat 2
Bass Clarinet B2
Bass Clarinet C3
Bass Clarinet D-flat 3
Bass Clarinet D3
Bass Clarinet E-flat 3
Bass Clarinet E3
Bass Clarinet F3
Bass Clarinet G-flat 3
Bass Clarinet G3
Bass Clarinet A-flat 3
Bass Clarinet A3
Bass Clarinet B-flat 3
Bass Clarinet B3
Bass Clarinet C4
Bass Clarinet D-flat 4
Bass Clarinet D4
Bass Clarinet E-flat 4
Bass Clarinet E4
Bass Clarinet F4
Bass Clarinet G-flat 4
Bass Clarinet G4
Bass Clarinet A-flat 4
Bass Clarinet A4
Bass Clarinet B-flat 4
Bass Clarinet B4
Bass Clarinet C5
Bass Clarinet D-flat 5
Bass Clarinet D5
Bass Clarinet E-flat 5
Bass Clarinet E5
Bass Clarinet F5
Bass Clarinet G-flat 5
Bass Clarinet G5
Bass Clarinet A-flat 5
Bass Clarinet A5
Bass Clarinet B-flat 5
Bassoon B-flat 1
Bassoon B1
Bassoon C2
Bassoon D-flat 2
Bassoon D2
Bassoon E-flat 2
Bassoon E2
Bassoon F2
Bassoon G-flat 2
Bassoon G2
Bassoon A-flat 2
Bassoon A2
Bassoon B-flat 2
Bassoon B2
Bassoon C3
Bassoon D-flat 3
Bassoon D3
Bassoon E-flat 3
Bassoon E3
Bassoon F3
Bassoon G-flat 3
Bassoon G3
Bassoon A-flat 3
Bassoon A3
Bassoon B-flat 3
Bassoon B3
Bassoon C4
Bassoon D-flat 4
Bassoon D4
Bassoon E-flat 4
Bassoon E4
Bassoon F4
Bassoon G-flat 4
Bassoon G4
Bassoon A-flat 4
Bassoon A4
Bassoon B-flat 4
Bassoon B4
Cello C2
Cello D-flat 2
Cello D2
Cello E-flat 2
Cello E2
Cello F2
Cello G-flat 2
Cello G2
Cello A-flat 2
Cello A2
Cello B-flat 2
Cello B2
Cello C3
Cello D-flat 3
Cello D3
Cello E-flat 3
Cello E3
Cello F3
Cello G-flat 3
Cello G3
Cello A-flat 3
Cello A3
Cello B-flat 3
Cello B3
Cello C4
Cello D-flat 4
Cello D4
Cello E-flat 4
Cello E4
Cello F4
Cello G-flat 4
Cello G4
Cello A-flat 4
Cello A4
Cello B-flat 4
Cello B4
Cello C5
Cello D-flat 5
Cello D5
Cello E-flat 5
Cello E5
Cello F5
Cello G-flat 5
Cello G5
Cello A-flat 5
Cello A5
Cello B-flat 5
Cello B5
Cello C6
Clarinet D3
Clarinet E-flat 3
Clarinet E3
Clarinet F3
Clarinet G-flat 3
Clarinet G3
Clarinet A-flat 3
Clarinet A3
Clarinet B-flat 3
Clarinet B3
Clarinet C4
Clarinet D-flat 4
Clarinet D4
Clarinet E-flat 4
Clarinet E4
Clarinet F4
Clarinet G-flat 4
Clarinet G4
Clarinet A-flat 4
Clarinet A4
Clarinet B-flat 4
Clarinet B4
Clarinet C5
Clarinet D-flat 5
Clarinet D5
Clarinet E-flat 5
Clarinet E5
Clarinet F5
Clarinet G-flat 5
Clarinet G5
Clarinet A-flat 5
Clarinet A5
Clarinet B-flat 5
Clarinet B5
Clarinet C6
Clarinet D-flat 6
Clarinet D6
Clarinet E-flat 6
Clarinet E6
Clarinet F6
Clarinet G-flat 6
Clarinet G6
Clarinet A-flat 6
Clarinet A6
Clarinet B-flat 6
Clarinet B6
Clarinet C7
Contra Bassoon B0
Contra Bassoon C1
Contra Bassoon D-flat 1
Contra Bassoon D1
Contra Bassoon E-flat 1
Contra Bassoon E1
Contra Bassoon F1
Contra Bassoon G-flat 1
Contra Bassoon G1
Contra Bassoon A-flat 1
Contra Bassoon A1
Contra Bassoon B-flat 1
Contra Bassoon B1
Contra Bassoon C2
Contra Bassoon D-flat 2
Contra Bassoon D2
Contra Bassoon E-flat 2
Contra Bassoon E2
Contra Bassoon F2
Contra Bassoon G-flat 2
Contra Bassoon G2
Contra Bassoon A-flat 2
Contra Bassoon A2
Contra Bassoon B-flat 2
Contra Bassoon B2
Contra Bassoon C3
Contra Bassoon D-flat 3
Contra Bassoon D3
Contra Bassoon E-flat 3
Contra Bassoon E3
Contra Bassoon F3
Contra Bassoon G-flat 3
Contra Bassoon G3
Contra Bassoon A-flat 3
Contra Bassoon A3
Contra Bassoon B-flat 3
Contra Bassoon B3
Contra Bassoon C4
Contra Bassoon D-flat 4
Double Bass C1
Double Bass D-flat 1
Double Bass D1
Double Bass E-flat 1
Double Bass E1
Double Bass F1
Double Bass G-flat 1
Double Bass G1
Double Bass A-flat 1
Double Bass A1
Double Bass B-flat 1
Double Bass B1
Double Bass C2
Double Bass D-flat 2
Double Bass D2
Double Bass E-flat 2
Double Bass E2
Double Bass F2
Double Bass G-flat 2
Double Bass G2
Double Bass A-flat 2
Double Bass A2
Double Bass B-flat 2
Double Bass B2
Double Bass C3
Double Bass D-flat 3
Double Bass D3
Double Bass E-flat 3
Double Bass E3
Double Bass F3
Double Bass G-flat 3
Double Bass G3
Double Bass A-flat 3
Double Bass A3
Double Bass B-flat 3
Double Bass B3
Double Bass C4
Double Bass D-flat 4
Double Bass D4
Double Bass E-flat 4
Double Bass E4
Double Bass F4
Double Bass G-flat 4
Double Bass G4
English Horn E3
English Horn F3
English Horn G-flat 3
English Horn G3
English Horn A-flat 3
English Horn A3
English Horn B-flat 3
English Horn B3
English Horn C4
English Horn D-flat 4
English Horn D4
English Horn E-flat 4
English Horn E4
English Horn F4
English Horn G-flat 4
English Horn G4
English Horn A-flat 4
English Horn A4
English Horn B-flat 4
English Horn B4
English Horn C5
English Horn D-flat 5
English Horn D5
English Horn E-flat 5
English Horn E5
English Horn F5
English Horn G-flat 5
English Horn G5
English Horn A-flat 5
English Horn A5
English Horn B-flat 5
English Horn B5
Flute C4
Flute D-flat 4
Flute D4
Flute E-flat 4
Flute E4
Flute F4
Flute G-flat 4
Flute G4
Flute A-flat 4
Flute A4
Flute B-flat 4
Flute B4
Flute C5
Flute D-flat 5
Flute D5
Flute E-flat 5
Flute E5
Flute F5
Flute G-flat 5
Flute G5
Flute A-flat 5
Flute A5
Flute B-flat 5
Flute B5
Flute C6
Flute D-flat 6
Flute D6
Flute E-flat 6
Flute E6
Flute F6
Flute G-flat 6
Flute G6
Flute A-flat 6
Flute A6
Flute B-flat 6
Flute B6
Flute C7
Flute D-flat 7
Flute D7
French Horn B-flat 1
French Horn B1
French Horn C2
French Horn D-flat 2
French Horn D2
French Horn E-flat 2
French Horn E2
French Horn F2
French Horn G-flat 2 (sorry, G-flat-2 missing)
French Horn G2
French Horn A-flat 2
French Horn A2
French Horn B-flat 2
French Horn B2
French Horn C3
French Horn D-flat 3
French Horn D3
French Horn E-flat 3
French Horn E3
French Horn F3
French Horn G-flat 3
French Horn G3
French Horn A-flat 3
French Horn A3
French Horn B-flat 3
French Horn B3
French Horn C4
French Horn D-flat 4
French Horn D4
French Horn E-flat 4
French Horn E4
French Horn F4
French Horn G-flat 4
French Horn G4
French Horn A-flat 4
French Horn A4
French Horn B-flat 4
French Horn B4
French Horn C5
French Horn D-flat 5
French Horn D5
French Horn E-flat 5
French Horn E5
French Horn F5
Guitar E2
Guitar F2
Guitar G-flat 2
Guitar G2
Guitar A-flat 2
Guitar A2
Guitar B-flat 2
Guitar B2
Guitar C3
Guitar D-flat 3
Guitar D3
Guitar E-flat 3
Guitar E3
Guitar F3
Guitar G-flat 3
Guitar G3
Guitar A-flat 3
Guitar A3
Guitar B-flat 3
Guitar B3
Guitar C4
Guitar D-flat 4
Guitar D4
Guitar E-flat 4
Guitar E4
Guitar F4
Guitar G-flat 4
Guitar G4
Guitar A-flat 4
Guitar A4
Guitar B-flat 4
Guitar B4
Guitar C5
Guitar D-flat 5
Guitar D5
Guitar E-flat 5
Guitar E5
Guitar F5
Guitar G-flat 5
Guitar G5
Guitar A-flat 5
Guitar A5
Guitar B-flat 5
Oboe B3
Oboe C4
Oboe D-flat 4
Oboe D4
Oboe E-flat 4
Oboe E4
Oboe F4
Oboe G-flat 4
Oboe G4
Oboe A-flat 4
Oboe A4
Oboe B-flat 4
Oboe B4
Oboe C5
Oboe D-flat 5
Oboe D5
Oboe E-flat 5
Oboe E5
Oboe F5
Oboe G-flat 5
Oboe G5
Oboe A-flat 5
Oboe A5
Oboe B-flat 5
Oboe B5
Oboe C6
Oboe D-flat 6
Oboe D6
Oboe E-flat 6
Oboe E6
Oboe F6
Oboe G-flat 6
Oboe G6
Oboe A-flat 6
Trombone E2
Trombone F2
Trombone G-flat 2
Trombone G2
Trombone A-flat 2
Trombone A2
Trombone B-flat 2
Trombone B2
Trombone C3
Trombone D-flat 3
Trombone D3
Trombone E-flat 3
Trombone E3
Trombone F3
Trombone G-flat 3
Trombone G3
Trombone A-flat 3
Trombone A3
Trombone B-flat 3
Trombone B3
Trombone C4
Trombone D-flat 4
Trombone D4
Trombone E-flat 4
Trombone E4
Trombone F4
Trombone G-flat 4
Trombone G4
Trombone A-flat 4
Trombone A4
Trombone B-flat 4
Trombone B4
Trombone C5
Trombone D-flat 5
Trombone D5
Trombone E-flat 5
Trombone E5
Trumpet E3
Trumpet F3
Trumpet G3
Trumpet A-flat 3
Trumpet A3
Trumpet B-flat 3
Trumpet B3
Trumpet C4
Trumpet D-flat 4
Trumpet D4
Trumpet E-flat 4
Trumpet E4
Trumpet F4
Trumpet G4
Trumpet A-flat 4
Trumpet A4
Trumpet B-flat 4
Trumpet B4
Trumpet C5
Trumpet D-flat 5
Trumpet D5
Trumpet E-flat 5
Trumpet E5
Trumpet F5
Trumpet G5
Trumpet A-flat 5
Trumpet A5
Trumpet B-flat 5
Trumpet B5
Trumpet C6
Trumpet D-flat 6
Trumpet D6
Trumpet E-flat 6
Trumpet E6
Tuba F1
Tuba G-flat 1
Tuba G1
Tuba A-flat 1
Tuba A1
Tuba B-flat 1
Tuba B1
Tuba C2
Tuba D-flat 2
Tuba D2
Tuba E-flat 2
Tuba E2
Tuba F2
Tuba G-flat 2
Tuba G2
Tuba A-flat 2
Tuba A2
Tuba B-flat 2
Tuba B2
Tuba C3
Tuba D-flat 3
Tuba D3
Tuba E-flat 3
Tuba E3
Tuba F3
Tuba G-flat 3
Tuba G3
Tuba A-flat 3
Tuba A3
Tuba B-flat 3
Tuba B3
Tuba C4
Tuba D-flat 4
Tuba D4
Tuba E-flat 4
Tuba E4
Tuba F4
Viola C3
Viola D-flat 3
Viola D3
Viola E-flat 3
Viola E3
Viola F3
Viola G-flat 3
Viola G3
Viola A-flat 3
Viola A3
Viola B-flat 3
Viola B3
Viola C4
Viola D-flat 4
Viola D4
Viola E-flat 4
Viola E4
Viola F4
Viola G-flat 4
Viola G4
Viola A-flat 4
Viola A4
Viola B-flat 4
Viola B4
Viola C5
Viola D-flat 5
Viola D5
Viola E-flat 5
Viola E5
Viola F5
Viola G-flat 5
Viola G5
Viola A-flat 5
Viola A5
Viola B-flat 5
Viola B5
Viola C6
Viola D-flat 6
Viola D6
Viola E-flat 6
Viola E6
Viola F6
Viola G-flat 6
Viola G6
Viola A-flat 6
Viola A6
Viola B-flat 6
Viola B6
Viola C7
Violin G3
Violin A-flat 3
Violin A3
Violin B-flat 3
Violin B3
Violin C4
Violin D-flat 4
Violin D4
Violin E-flat 4
Violin E4
Violin F4
Violin G-flat 4
Violin G4
Violin A-flat 4
Violin A4
Violin B-flat 4
Violin B4
Violin C5
Violin D-flat 5
Violin D5
Violin E-flat 5
Violin E5
Violin F5
Violin G-flat 5
Violin G5
Violin A-flat 5
Violin A5
Violin B-flat 5
Violin B5
Violin C6
Violin D-flat 6
Violin D6
Violin E-flat 6
Violin E6
Violin F6
Violin G-flat 6
Violin G6
Violin A-flat 6
Violin A6
Violin B-flat 6
Violin B6
Violin C7
Violin D-flat 7
Violin D7
Violin E-flat 7
Violin E7
Violin F7
Violin G-flat 7
Violin G7
Female Vocal Soprano Bb3
Female Vocal Soprano C4
Female Vocal Soprano D4
Female Vocal Soprano Eb4
Female Vocal Soprano F4
Female Vocal Soprano G4
Female Vocal Soprano A4
Female Vocal Soprano Bb4
Female Vocal Soprano C5
Female Vocal Soprano D5
Female Vocal Soprano Eb5
Female Vocal Soprano F5
Female Vocal Soprano G5
Female Vocal Soprano A5
Female Vocal Soprano Bb5
User-Data Input Selector
This selector allows you to choose from several User-Data input signals, and to process your selected input signal with the Fast Fourier Transform (FFT).
These User-Data input signals are provided as examples of real-world signals that are often processed by the FFT, including musical instrument sounds, human voices in audio recordings, and other signals.
© 2024 Sooeet.com All rights reserved
Number of data samples:
Number of Data Samples
This output field displays the number of data samples that where processed by the Fourier transform.
If you provided an input list of data samples, the number of data samples displayed here should coincide exactly with the length of your input list. If it does not coincide, it is because your input list contained blank elements, such as:
1, 2, 3, , 5, 6
Calculator Memory
You can save the result of this output field in the Calculator Memory, then use the saved result as input for other converters and calculators anywhere on the website. To save this result, click inside the output field, and then click the button entitled
Save
in the Calculator Memory.
See the
Calculator Memory
for more details.
TIP:
Precision
of the calculator is 15 significant digits, from left to right. Digits after the 15 left-most digits, are not significant.
Numeric Range
of the calculator is:
+1.797e+308 ... -1.797e+308
+1.000e-311 ... -1.000e-311
© 2024 Sooeet.com All rights reserved
FFT real output:
FFT
Real Output
When you press the
Get FFT Data
button, this output field displays the contents of the Real-
FFT
result, as a list of numbers separated by blank spaces.
Calculator Memory
You can save the result of this output field in the Calculator Memory, then use the saved result as input for other converters and calculators anywhere on the website. To save this result, click inside the output field, and then click the button entitled
Save
in the Calculator Memory.
See the
Calculator Memory
for more details.
TIP:
Precision
of the calculator is 15 significant digits, from left to right. Digits after the 15 left-most digits, are not significant.
Numeric Range
of the calculator is:
+1.797e+308 ... -1.797e+308
+1.000e-311 ... -1.000e-311
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FFT imaginary output:
FFT
Imaginary Output
When you press the
Get FFT Data
button, this output field displays the contents of the Imaginary-
FFT
result, as a list of numbers separated by blank spaces.
Calculator Memory
You can save the result of this output field in the Calculator Memory, then use the saved result as input for other converters and calculators anywhere on the website. To save this result, click inside the output field, and then click the button entitled
Save
in the Calculator Memory.
See the
Calculator Memory
for more details.
TIP:
Precision
of the calculator is 15 significant digits, from left to right. Digits after the 15 left-most digits, are not significant.
Numeric Range
of the calculator is:
+1.797e+308 ... -1.797e+308
+1.000e-311 ... -1.000e-311
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FFT/IFFT graph data output:
FFT/IFFT
Graph Data Output
When you press the
Get FFT Data
button, this output field displays the current
FFT/IFFT
graph data, as a list of numbers separated by blank spaces.
The graph data that is output depends on the selection in the
FFT Y-Axis Magnitude
menu, and on the mode set by the
FFT/Inv-FFT
button, and on the mode set by the
Fold ω / Un-Fold ω
button, and on whether
User-Data
or
Program-Data
is currently selected.
The
FFT/IFFT Graph Data Output
can be used to connect the output of one
FFT
operation to the input of a second
FFT
operation. Simply copy and paste the
FFT/IFFT Graph Data Output
to the
User Data
input. Then, in the
Sampling Rate
input field, enter the size of the output buffer from the first
FFT
, and finally, click the
User Data
button.
Cepstrum
Analysis
The technique of using the output of one
FFT
operation as the input of a second
FFT
operation is used for example in
Cepstrum
analysis.
Cepstrum
analysis or
Cepstral
analysis attempts to detect periodic signals that may occur in the frequency spectrum of a time-domain or spatial-domain signal, or specifically, in the
FFT
of a time-domain or spatial-domain signal.
Calculator Memory
You can save the result of this output field in the Calculator Memory, then use the saved result as input for other converters and calculators anywhere on the website. To save this result, click inside the output field, and then click the button entitled
Save
in the Calculator Memory.
See the
Calculator Memory
for more details.
TIP:
Precision
of the calculator is 15 significant digits, from left to right. Digits after the 15 left-most digits, are not significant.
Numeric Range
of the calculator is:
+1.797e+308 ... -1.797e+308
+1.000e-311 ... -1.000e-311
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y y y y
x y x y
– ω +
+ ω –
Control Buttons
User Data
Button
Click this button to perform the Fast Fourier Transform (
FFT
) using data from the
User Data
source. The
FFT
calculator will render a graph in the frequency domain, or in the time-domain, depending on which of those modes is currently active.
Program Data
Button
Click this button to perform the Fast Fourier Transform (
FFT
) using data from the
Program Data
source. The
FFT
calculator will render a graph in the frequency domain, or in the time-domain, depending on which of those modes is currently active.
Get FFT Data
Button
When you click the
Get FFT Data
button, the three
FFT
output fields display the current
FFT/IFFT
graph data, as lists of numbers separated by blank spaces. The three
FFT
output fields are:
FFT Real Output
FFT Imaginary Output
FFT/IFFT Graph Data Output
yyyy
Radio Button
Sets the data format for the
FFT
results that are retrieved when you click the
Get FFT Data
button. When this radio button is set, the
FFT
results that are retrieved contain y-values only. The x-values are implied.
xyxy
Radio Button
Sets the data format for the
FFT
results that are retrieved when you click the
Get FFT Data
button. When this radio button is set, the
FFT
results that are retrieved contain x-values and y-values as x y pairs.
– ω +
Radio Button
Sets the data format for the
FFT
results that are retrieved when you click the
Get FFT Data
button. When this radio button is set, the
FFT
results that are retrieved are organized so that the most-negative frequencies are given first, then the least-negative frequencies are given, then the zero frequency is given, then the least-positive frequencies are given, and finally the most-positive frequencies are given.
+ ω –
Radio Button
Sets the data format for the
FFT
results that are retrieved when you click the
Get FFT Data
button. When this radio button is set, the
FFT
results that are retrieved are organized so that the zero frequency is given first, then the least-positive frequencies are given, then the most-positive frequencies are given, then the most-negative frequencies are given, and finally the least-negative frequencies are given.
This is the natural format of the
FFT
results.
Screen Shot
Button
This button allows you to save a JPEG image of the currenly displayed graph, to your computer.
Fold ω / Un-Fold ω
Button
This button folds or unfolds 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.
Show Points / Hide Points
Button
This button displays the individual data samples on the time-domain graph, and on the frequency-domain graph. When the button is clicked, it changes to the Hide Points mode. Clicking the button in the Hide Points mode, hides the individual data samples.
Inv-FFT / FFT
Button
This button runs the
FFT
(forward Fast Fourier Transform), or the
IFFT
(inverse Fast Fourier Transform), on the input data.
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How to use the FFT calculator:
How to Use the
FFT
Calculator
The Fourier Transform Calculator performs
FFT
and
IFFT
transforms of input data, and displays time-domain, spatial-domain and frequency-domain representations of that data in graphical form. The Fourier Transform Calculator accepts three different sources of data.
Source 1:
Program Data
The first source of data for the Fourier Transform Calculator is the time-domain
Program Data
source. A large number of
Program Data
objects are programmed into the calculator. Any
Program Data
object may be loaded by the user whenever that data object is required.
The menu entitled
"Program-Data"
provides the list of all the available time-domain
Program Data
objects. To use this source of data, select one of the
Program Data
objects from that menu.
The Fourier Transform Calculator will automatically process the selected
Program Data
object via the
FFT
, and display the frequency-domain spectrum, or the time-domain function, for that
Program Data
object.
The menu entitled
"FFT Number of Samples"
applies only to the
Program Data
objects. Use that menu to choose the number of samples that are taken from the
Program Data
object currently selected in the menu entitled
"Program-Data"
.
Source 2:
User Data
The second source of data for the Fourier Transform Calculator is
User Data
. This is simply a list of time-domain data samples that the user provides to the calculator for processing via the
FFT
.
To employ the
User Data
source, enter a list of time-domain data samples in the first input box, then enter the sampling rate of that data in the second input box, and click the
User Data
button.
The Fourier Transform Calculator will automatically process your input data via the
FFT
, and display the frequency-domain spectrum, or the time-domain function, for the given
User 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, 1024, 2048, 4096, 8192, 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.
Source 3:
User Data Examples
The third source of data for the Fourier Transform Calculator is
User-Data Examples
. Many examples of User-Data are provided in the selector box entitled "User-Data Examples".
To employ the
User Data Examples
source, simply select one of the items in the selector box entitled "User-Data Examples".
The Fourier Transform Calculator will automatically process your selected User-Data example via the
FFT
, and display the frequency-domain spectrum, or the time-domain function, for your selected
User-Data Example
.
How to Change the Source of Data
To change to the
User Data
source, simply enter a list of data samples in the first input box, enter the sampling rate of that data in the second input box, and click the
User Data
button.
To change to the
Program Data
source, simply click the
Program Data
button, or select a new
Program Data
object from the menu entitled
"Program-Data"
.
To change to the
User Data Examples
source, simply select one of the items from the selector box entitled "User-Data Examples".
The
FFT
calculator will operate on the newly selected data source until you change to another data source.
Graphing Mode
The
FFT
calculator uses Sooeet Graph v 1.5F to plot the frequency-domain spectrum of the Fourier Transformed output, and the time-domain representation of the input data.
Sooeet Graph v 1.5F is preset to "Automatic" grid mode. In that mode the x and y grids are automatically scaled to fully contain the frequency-domain spectrum of the Fourier Transformed output, or the time-domain representation of the input data.
Decimal and Scientific Notation
The
FFT
calculator automatically outputs in Decimal Notation, or in Scientific Notation, depending on the magnitude of the input data, and depending on the magnitude of the Fourier Transform output in the frequency-domain.
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FFT frequency domain output:
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 spectral 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 in cycles per unit length. However, the actual units of the x-axis depend on the nature of the input data.
For example, if the input data represents an electrical signal or a mechanical signal, the units of the x-axis are cycles per second, or Hertz, but if the input data represents a row or a column from a 2-dimensional image, then the units of the x-axis are cycles per unit of length.
Y-Axis of the Spectral Graph
The y-axis represents the magnitude of the frequency components in the spectral output of the FFT. The magnitude of the frequency components can be expressed in either of two ways:
Amplitude-Units-Peak
Decibels of Amplitude-Units-Peak
Amplitude-Units-Peak (Apk)
This is a direct linear representation of the FFT output magnitude. Use Apk when you need to see the actual magnitude of large spectral components. However, when using the Apk scale keep in mind that small spectral components are likely burried in the presence of much larger spectral components. Amplitude-Units-Peak (Apk) is calculated as follows:
Apk = Sqrt( Re-FFT()^2 + Im-FFT()^2 ) / N
Decibels of Amplitude-Units-Peak (dBV)
This is a logarithmic (log base-10) measure of the ratio of the FFT output Amplitude-Units-Peak to a reference value of 1-Amplitude-Unit. Use dBV when you need to easily see small spectral components in the presence of large spectral components. dBV is calculated as follows:
dBV = 20 log( Apk / 1-Amplitude-Unit )
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FFT y-axis magnitude:
FFT
Frequency Domain Y-Axis Magnitude
The Y-Axis Magnitude Options
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:
FFT Y-Axis Magnitude
Menu
This menu has the following options:
Apk = Sqrt( R^2 + I^2 ) / N
Square root of the sum of the squared Real component (R) of the
FFT
result and the squared Imaginary component (I) of the
FFT
result, divided by the number of samples in the
FFT
input buffer (N).
Amplitude-Units-Peak (Apk) is a direct linear representation of the
FFT
output magnitude. Use Apk when you need to see the actual magnitude of large spectral components. However, when using the Apk scale keep in mind that small spectral components are likely burried in the presence of much larger spectral components.
Sqrt( R^2 + I^2 )
Square root of the sum of the squared Real component (R) of the
FFT
result and the squared Imaginary component (I) of the
FFT
result.
Real^2 + Imag^2
The sum of the squared Real component (R) of the
FFT
result and the squared Imaginary component (I) of the
FFT
result.
Real^2
The square of the Real component of the
FFT
result.
Imaginary^2
The square of the Imaginary component of the
FFT
result.
Real
The Real component of the
FFT
result.
Imaginary
The Imaginary component of the
FFT
result.
dBV = 20 log( Sqrt( R^2 + I^2 ) / N )
The decibel of the square root of the sum of the squared Real component (R) of the
FFT
result and the squared Imaginary component (I) of the
FFT
result, divided by the number of samples in the
FFT
input buffer (N).
This is a logarithmic measure of the ratio of the FFT output Amplitude-Units-Peak (Apk) to a reference value of 1-Amplitude-Unit. Use dBV when you need to easily see small spectral components in the presence of large spectral components.
20 log( Sqrt( R^2 + I^2 ) )
The decibel of the square root of the sum of the squared Real component (R) of the
FFT
result and the squared Imaginary component (I) of the
FFT
result.
20 log( R^2 + I^2 )
The decibel of the sum of the squared Real component (R) of the
FFT
result and the squared Imaginary component (I) of the
FFT
result.
20 log( Real^2 )
The decibels of the squared Real component of the
FFT
result.
20 log( Imag^2 )
The decibels of the squared Imaginary component of the
FFT
result.
Norm(20 log( ... ))
Similar to the above decibel values, but normalized so the peak value is at 0 dB.
Phase( FFT() ) in radians
Phase spectrum in radians, derived from the FFT output.
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FFT window functions:
The Window Functions
FFT
Window Functions
1
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 non-periodic finite-length sequence of input data.
To apply a window function, simply select one of the window functions from the menu entitled
"FFT Window Function"
. The selected window function is automatically applied to the sample data prior to the
FFT
.
Program-Data
Input time-domain data for the
FFT
can be supplied as a Sample List of
User Data
, or by one of the
Program Data
objects available in the menu entitled
"Program-Data"
. That menu contains an assortment of preset
Program Data
objects, including all of the window functions.
The window functions are provided in the menu entitled
"Program-Data"
, so that users may analyze the time-domain and frequency-domain characteristics of the window functions.
However, to pre-multiply a window function with
FFT
input data, use the menu entitled
"FFT Window Function"
to select a window function of your choice, rather than the menu entitled
"Program-Data"
.
The
FFT
Window Functions
The following window functions are provided:
Rectangle, Triangle, Hann (aka Hanning), Raised Cosine (1-4), 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, 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).
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Sampling rate for
User Data
:
Sampling Rate of
User Data
Sampling Rate
When providing
User Data
to the
FFT
, you must also enter the sampling rate of that data, in Hertz, (i.e. samples per second.) This only applies to
User Data
. The sampling rate for
Program Data
is set automatically.
Sampling Rate Format
The sampling rate value that you enter must be in one of the following formats:
Integer:
8820
Decimal:
44100.9
Scientific Notation:
1.102536e04
Nyquist–Shannon Sampling Theorem
The Nyquist–Shannon theorem is one of the principal elements of information theory. The theorem states that a bandlimited analog signal can be completely reconstructed via the inverse Fourier transform, only if the signal is sampled at a rate that exceeds
2F
, where
F
is the highest frequency contained in the original signal.
Therefore, when sampling a bandlimited analog signal intended for processing via the
FFT
, the sampling rate should exceed twice the maximum frequency that you wish to resolve with the
FFT
.
For example, to capture a 3 Kilohertz frequency component in the original analog signal, the analog signal must be sampled at a rate that exceeds 6 Kilohertz.
Limitations of the Fast Fourier Transform
For best results, the number of samples in the input data list should be a power of 2, such as 1024, 4096, 32768, upto a maximum of:
2^18 = 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.
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FFT parametric controls:
FFT
Parametric Controls
Slider Controls
Some of the
Program Data
objects allow the user to control phase, frequency, pulse-width, and other parameters that drive the
Program Data
object.
One of the parametric controls is the Slider, which allows the user to modify one parameter, by clicking and dragging the Slider button. The Slider provides a quick way to modify the parameter it controls, but it does not provide fine control over that parameter.
Other Parametric Controls
For precise control over frequency and other parameters, the following buttons are provided:
F/2
Button
This button lowers the current frequency, dividing that frequency by 2. For example, if the current frequency is 120 Hz, clicking this button lowers the frequency to 60 Hz. Clicking and holding this button, causes this action to be repeated until the button is released.
2F
Button
This button raises the current frequency, multiplying that frequency by 2. For example, if the current frequency is 150 Hz, clicking this button raises the frequency to 300 Hz. Clicking and holding this button, causes this action to be repeated until the button is released.
-0.1
Button
This button lowers the current frequency by subtracting 0.1 Hz from that frequency. For example, if the current frequency is 60 Hz, clicking this button lowers the frequency to 59.9 Hz. Clicking and holding this button, causes this action to be repeated until the button is released.
+0.1
Button
This button raises the current frequency by adding 0.1 Hz to that frequency. For example, if the current frequency is 60 Hz, clicking this button raises the frequency to 60.1 Hz. Clicking and holding this button, causes this action to be repeated until the button is released.
-0.01
Button
This button lowers the current frequency by subtracting 0.01 Hz from that frequency. For example, if the current frequency is 60 Hz, clicking this button lowers the frequency to 59.99 Hz. Clicking and holding this button, causes this action to be repeated until the button is released.
+0.01
Button
This button raises the current frequency by adding 0.01 Hz to that frequency. For example, if the current frequency is 60 Hz, clicking this button raises the frequency to 60.01 Hz. Clicking and holding this button, causes this action to be repeated until the button is released.
FLR
Button
This button Floors the current frequency to the nearest lower integer value. For example, if the current frequency is 60.759 Hz, clicking this button Floors the frequency to 60.000 Hz. Clicking and holding this button, causes this action to be repeated until the button is released.
Arrow Keys on the Keyboard
The Left and Right arrow keys on the keyboard can also be used to lower and raise the current frequency. To use these buttons, first click the frequency slider button to select it, then press the Left or Right arrow keys on the keyboard. Holding down an arrow key, causes this action to be repeated until the arrow key is released.
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FFT sound:
FFT
Sound
WARNING: RISK of HEARING DAMAGE
The
FFT
calculator generates sounds which may cause permanent hearing damage in humans and animals who are exposed to those sounds at high amplification, and/or for extended periods of time.
Do not expose yourself, or other people, or animals, to the
FFT
calculator sounds, at high amplification, or for extended periods of time even at low levels of amplification.
Do not expose yourself, or other people, or animals, to the
FFT
calculator sounds, via headphones, at high amplification, or for extended periods of time even at low levels of amplification.
Fourier and Sound Priciples
The forward Fourier Transform, implemented by this calculator as the
FFT
, produces a frequency-domain power spectrum. The inverse Fourier Transform, implemented by this calculator as the
IFFT
, produces a time-domain or spatial-domain signal or function, that can be interpreted as a time-varying voltage.
If we interpret the time-domain or spatial-domain function or signal that is produced by the
IFFT
, as a time-varying voltage, then we can use that function or signal to drive an audio amplifier, and we can use the output of the amplifier to drive an electro-mechanical audio transducer, otherwise known as a "Speaker".
Sound in air is produced by compression and rarefaction of air molecules, by a mechanical agent. In an electronic audio system, the mechanical agent is an electro-mechanical audio transducer. Vibrations of the transducer produce a sound wave that humans can perceive, when the frequency of vibration is in the range of 15 Hz to 20 KHz.
The
FFT
calculator uses the audio system of the computer on which it is running, to generate audio signals in the frequency range of 0.001 Hz to well above the 20 KHz upper frequency limit of human hearing.
Generally, audio frequencies below 15 Hz are considered infrasonic, meaning that humans generally cannot hear such low frequencies, and frequencies above 20 KHz are considered ultrasonic, meaning that humans generally cannot hear such high frequencies.
Sound and the Fourier Series
Perhaps the best way to illustrate the use of sound in the
FFT
calculator, is by way of an example. The menu entitled
"Program-Data"
contains an item entitled
"Synthesis - Square Wave"
. Please choose that item from that menu.
You should see three new Sliders and several new Buttons. These controls are used to manage the synthesis of a square wave from the sum of a series of trigonometric sine waves.
The series of trigonometric sine waves that the
FFT
calculator uses to synthesize a square wave, is a so-called
Fourier Series
, a mathematical series directly related to the development of the continuous and discrete Fourier Transforms.
Move the Slider entitled
"Fund. (Square Wave)"
. This Slider controls the so-called
Fundamental Frequency
of the synthesized square wave. Choose a fundamental frequency of about 16 Hz.
Now move the Slider entitled
"Harm. (Square Wave)"
. This Slider controls the number of so-called
Odd Harmonics
that are summed to produce the synthesized square wave. Choose 10 harmonics.
Now click the button entitled
Play
. If the audio system on your computer is working properly, you should hear a low pitched hum, which contains some higher pitched sounds within it.
Now click the button entitled
Inv-FFT / FFT
. You should see the graphical representation of the audio signal in the Spatial-Domain, and the frequency components that make up that audio signal in the Frequency-Domain.
To stop the sound click the button entitled
Stop
.
Limitations of the
FFT
Sound System
Due to the limitations of the
FFT
sound system, the sound buffer that is used for
FFT
sound output always generates 8192 sound samples at a sampling rate of 44.1 KHz, regardless of the setting in the menu entitled
"FFT Number of Samples"
.
Due to this limitation, sound frequencies above 8.192 KHz will be distorted to the degree that they exceed 8.192 KHz. However, in practical use, this limitation is of little consequence, because most audio amplifiers and speakers cannot accurately reproduce high frequencies.
This limitation is a factor if the sound output from the
FFT
calculator is reproduced over a high quality audio system, which includes a high quality preamplifier, a high quality power amplifier, and high quality speakers.
© 2024 Sooeet.com All rights reserved
FFT additive synthesis:
FFT
Additive Synthesis
Fourier and Additive Synthesis
As explained in the "FFT Sound" help topic, using Square Wave Synthesis via the Fourier sine series it is possible to synthesize an approximation of a square wave by summing a specific series of sine waves.
However, more generally, it is possible to synthesize an approximation of any periodic function by summing a carefully crafted finite series of sine waves and/or cosine waves.
The menu entitled
"Program-Data"
contains an item entitled
"Sinusoid Component 1"
. Please choose that item from that menu.
You should see four new Sliders and several new Buttons. These controls are used to manage the synthesis of an arbitrary function from the sum of a series of trigonometric sine and cosine waves.
The item entitled
"Sinusoid Component 1"
is the first of 16 sinusoid components that can be used to approximate many arbitrary functions by means of Fourier Synthesis.
The button entitled
Sinusoid / Component
allows you to see and hear either the entire synthesized sinusoid, or the individual component sine or cosine wave.
The item entitled
"Synthesis - Sinusoid"
in the menu entitled
"Program-Data"
, is the synthesized master-sinusoid, which is composed of upto 16 components.
The item entitled
"Synthesis - Sinusoid"
also allows the addition of random white Gaussian noise to the synthesized function, via the slider entitled
"Noise Level (Sinusoid)"
.
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