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Transforms
dspxfrm3
The FFT block computes the fast Fourier transform (FFT) of each row of a sample-based 1-by-P input vector, u, or across the first dimension (P) of an N-D input array, u. When you select the Inherit FFT length from input dimensions check box, P must be an integer power of two, and the FFT length, M, is set equal to P. If you do not select the check box, P can be any length, and the value of the FFT length parameter must be a positive integer power of two. For user-specified FFT lengths, when M does not equal P, zero padding or modulo-M data wrapping occurs before the FFT operation, as per Orfanidis [1]:
y = fft(u,M) % P ≤ M
y(:,l) = fft(datawrap(u(:,l),M)) % P > M; l = 1,...,N
When the input length, P, is greater than the FFT length, M, you may see magnitude increases in your FFT output. These magnitude increases occur because the FFT block uses modulo-M data wrapping to preserve all available input samples.
To avoid such magnitude increases, you can truncate the length of your input sample, P, to the FFT length, M. To do so, place a Pad block before the FFT block in your model.
The kth entry of the lth output channel, y(k, l), equals the kth point of the M-point discrete Fourier transform (DFT) of the lth input channel:
This block supports real and complex floating-point and fixed-point inputs.
The following table describes valid inputs to the FFT block, their corresponding outputs, and the dimension along which the block computes the DFT.
Valid inputs to the FFT block. Inputs can be real- or complex-valued, and must be in linear order.
The dimension along which the block computes the DFT.
The corresponding block output characteristics. The output port rate must equal the input port rate.
| Valid Block Inputs | Dimension Along Which Block Computes DFT | Corresponding Block Output Characteristics |
|---|---|---|
N-D array | First dimension |
|
Sample-based 1-by-P row vector | Row |
|
Unoriented length-P 1-D vector | Vector | Unoriented, length-M, complex-valued 1-D output vector containing M-point DFT of input in linear or bit-reversed order. |
The Twiddle factor computation parameter
determines how the block computes the necessary sine and cosine terms
to calculate the term
,
shown in the first equation of this block reference page.
The block only supports Table lookup mode for fixed-point signals.
The Twiddle factor computation parameter has two settings, each with its advantages and disadvantages, as described in the following table.
| Twiddle Factor Computation Parameter Setting | Sine and Cosine Computation Method | Effect on Block Performance |
|---|---|---|
Table lookup | The block computes and stores the trigonometric values before the simulation starts and retrieves them during the simulation. When you generate code from the block, the processor running the generated code stores the trigonometric values computed by the block, and retrieves the values during code execution. | The block usually runs much more quickly, but requires extra memory for storing the precomputed trigonometric values. You can optimize the table for memory consumption or speed, as described in Optimizing the Table of Trigonometric Values. |
Trigonometric fcn | The block computes sine and cosine values during the simulation. When you generate code from the block, the processor running the generated code computes the sine and cosine values while the code runs. | The block usually runs more slowly, but does not need extra data memory. For code generation, the block requires a support library to emulate the trigonometric functions, increasing the size of the generated code. |
When you set the Twiddle factor computation parameter to Table lookup, you also need to set the Optimize table for parameter.
This parameter optimizes the table of trigonometric values for speed or memory by varying the number of table entries as summarized in the following table.
| Optimize Table for Parameter Setting | Number of Table Entries for N-Point FFT | Memory Required for Single-Precision, 512-Point FFT |
|---|---|---|
Speed | 3N/4 — floating point N — fixed point |
|
Memory | N/4 — floating point Not supported for fixed point |
|
You can set the Output in bit-reversed order parameter to specify the ordering of the column elements of the block output. If you select the Output in bit-reversed order check box, the output appears in bit-reversed order. If you clear the Output in bit-reversed order check box, the output appears in linear order.
Note The FFT block calculates its output in bit-reversed order. Linearly ordering the FFT block output requires an extra bit-reversal operation. Thus, in many situations, you can increase the speed of the FFT block by selecting the Output in bit-reversed order check box. |
For more information ordering of the output, see Linear and Bit-Reversed Output Order.
Depending on whether the block's input is real- or complex-valued and whether you want the output in linear or bit-reversed order, the block uses one or more of the following algorithms as summarized in the following table:
Bit-reversal operation
Double-signal algorithm
Half-length algorithm
Radix-2 decimation-in-time (DIT) algorithm
Radix-2 decimation-in-frequency (DIF) algorithm
| Complexity of Input | Output Ordering | Algorithms Used for FFT Computation |
|---|---|---|
Complex | Linear | Bit-reversal operation and radix-2 DIT |
Complex | Bit-reversed | Radix-2 DIF |
Real | Linear | Bit-reversal operation and radix-2 DIT in conjunction with the half-length and double-signal algorithms |
Real | Bit-reversed | Radix-2 DIF in conjunction with the half-length and double-signal algorithms |
The efficiency of the FFT algorithm can be enhanced for real input signals by forming complex-valued sequences from the real-valued sequences prior to the computation of the DFT. When there are 2N+1 real input channels, the FFT block forms these complex-valued sequences by applying the double-signal algorithm to the first 2N input channels, and the half-length algorithm to the last odd-numbered channel.
For real input signals with fixed-point data types, it is possible to see different numerical results in the output of the last odd numbered channel, even when all input channels are identical. This numerical difference results from differences in the double-signal algorithm and the half-length algorithm.
You can eliminate this numerical difference in two ways:
Using full precision arithmetic for fixed-point input signals
Changing the input data type to floating point
For more information on the double-signal and half-length algorithms, see Proakis [2]. "Efficient Computation of the DFT of Two Real Sequences" on page 475 describes the double signal algorithm. "Efficient Computation of the DFT of a 2N-Point Real Sequence" on page 476 describes the half-length algorithm.
The following diagrams show the data types used within the FFT block for fixed-point signals. You can set the sine table, accumulator, product output, and output data types displayed in the diagrams in the FFT block dialog as discussed in Dialog Box.
The block first casts inputs to the FFT block to the output data type and stores them in the output buffer. Each butterfly stage then processes signals in the accumulator data type, and the block then casts the final output of the butterfly back into the output data type. The block multiplies in a twiddle factor before each butterfly stage in a decimation-in-time FFT, and after each butterfly stage in a decimation-in-frequency FFT.
The output of the multiplier appears in the accumulator data type because both of the inputs to the multiplier are complex. For details on the complex multiplication performed, see Multiplication Data Types.
See the section on Transforming Time-Domain Data into the Frequency Domain in the Signal Processing Blockset User's Guide.
The Main pane of the FFT block dialog appears as follows.
Specify the computation method of the term
, shown in the
first equation of this block reference page.
In Table lookup mode, the block computes and stores the sine and cosine values before the simulation starts.
In Trigonometric fcn mode, the block computes the sine and cosine values during the simulation. See Selecting the Twiddle Factor Computation Method.
This parameter must be set to Table lookup for fixed-point signals.
Select the optimization of the table of sine and cosine values for Speed or Memory. This parameter becomes available only when you set the Twiddle factor computation parameter to Table lookup. See Selecting the Twiddle Factor Computation Method.
This parameter must be set to Speed for fixed-point signals.
Designate the order of the output channel elements relative to the ordering of the input elements. When you select this check box, the output channel elements appear in bit-reversed order relative to the input ordering. Otherwise, the output column elements appear in linear order relative to the input ordering.
Linearly ordering the output requires extra data sorting manipulation, so in some situations it might be better to output in bit-reversed order.
When you select this parameter, the output of each butterfly of the FFT is divided by two. When you do not select this parameter, the block does not scale the output.
Select to inherit the FFT length from the input dimensions. When you select this parameter, the input length P must be a power of two. When you do not select this parameter, the FFT length parameter becomes available.
Specify a power-of-two FFT length. This parameter becomes available only when you do not select the Inherit FFT length from input dimensions parameter.
The Fixed-point pane of the FFT block dialog appears as follows.
Select the rounding mode for fixed-point operations. The sine table values do not obey this parameter; instead, they always round to Nearest.
Select the overflow mode for fixed-point operations. The sine table values do not obey this parameter; instead, they are always saturated.
Choose how you specify the word length of the values of the sine table. The fraction length of the sine table values always equals the word length minus one:
When you select Same word length as input, the word length of the sine table values match that of the input to the block.
When you select Specify word length, you can enter the word length of the sine table values, in bits.
The sine table values do not obey the Rounding mode and Overflow mode parameters; instead, they are always saturated and rounded to Nearest.
Use this parameter to specify how you would like to designate the product output word and fraction lengths. See Fixed-Point Data Types and Multiplication Data Types for illustrations depicting the use of the product output data type in this block:
When you select Inherit via internal rule, the block calculates the product output word length and fraction length automatically. For information about how the product output word and fraction lengths are calculated when an internal rule is used, see Inherit via Internal Rule.
When you select Same as input, these characteristics match those of the input to the block.
When you select Binary point scaling, you can enter the word length and the fraction length of the product output, in bits.
When you select Slope and bias scaling, you can enter the word length, in bits, and the slope of the product output. This block requires power-of-two slope and a bias of zero.
Use this parameter to specify how you would like to designate the accumulator word and fraction lengths. See Fixed-Point Data Types and Multiplication Data Types for illustrations depicting the use of the accumulator data type in this block:
When you select Inherit via internal rule, the block calculates the accumulator word length and fraction length automatically. For information about how the accumulator word and fraction lengths are calculated when an internal rule is used, see Inherit via Internal Rule.
When you select Same as product output, these characteristics match those of the product output.
When you select Same as input, these characteristics match those of the input to the block.
When you select Binary point scaling, you can enter the word length and the fraction length of the accumulator, in bits.
When you select Slope and bias scaling, you can enter the word length, in bits, and the slope of the accumulator. This block requires power-of-two slope and a bias of zero.
Choose how you specify the output word length and fraction length:
When you select Inherit via internal rule, the block calculates the output word length and fraction length automatically. The internal rule first calculates an ideal output word length and fraction length using the following equations:
Using these ideal results, the internal rule then selects word lengths and fraction lengths that are appropriate for your hardware. For more information, see Inherit via Internal Rule.
When you select Same as input, these characteristics match those of the input to the block.
When you select Binary point scaling, you can enter the word length and the fraction length of the output, in bits.
When you select Slope and bias scaling, you can enter the word length, in bits, and the slope of the output. This block requires power-of-two slope and a bias of zero.
Select this parameter to prevent any fixed-point scaling you specify in this block mask from being overridden by the autoscaling feature of the Fixed-Point Tool. See the fxptdlg reference page for more information.
[1] Orfanidis, S. J. Introduction to Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1996, p. 497.
[2] Proakis, John G. and Dimitris G. Manolakis. Digital Signal Processing, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 1996.
| Port | Supported Data Types |
|---|---|
Input |
|
Output |
|
| DCT | Signal Processing Blockset |
| IFFT | Signal Processing Blockset |
| Pad | Signal Processing Blockset |
| fft | MATLAB |
| ifft | MATLAB |
| bitrevorder | Signal Processing Toolbox |
 | Fast Block LMS Filter | Filter Realization Wizard |  |
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