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  4. Appendix 1: fft flowgraphs

0.3 Appendix 1: fft flowgraphs

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Flowgraphs of various radix-2 and 4 Cooley Tukey FFTs and Split Radix FFTs.

Signal flow graphs of cooley-tukey ffts

The following four figures are flow graphs for Radix-2 Cooley-Tukey FFTs. The first is a length-16, decimation-in-frequency Radix-2 FFT with the input data in order and output data scrambled. The first stage has 8 length-2 "butterflies" (which overlap in the figure) followed by 8 multiplications by powers of W which are called "twiddle factors". The second stage has 2 length-8 FFTs which are each calculated by 4 butterflies followed by 4 multiplies. The third stage has 4 length-4 FFTs, each calculated by 2 butterflies followed by 2 multiplies and the last stage is simply 8 butterflies followed by trivial multiplies by one. This flow graph should be compared with the index map in Polynomial Description of Signals , the polynomial decomposition in The DFT as Convolution or Filtering , and the program in Appendix 3 . In the program, the butterflies and twiddle factor multiplications are done together in the inner most loop. The outer most loop indexes through the stages. If the length of the FFT is a power of two, the number of stages is that power (log N).

The second figure below is a length-16, decimation-in-time FFT with the input data scrambled and output data in order. The first stage has 8 length-2 "butterflies" followed by 8 twiddle factors multiplications. The second stage has 4 length-4 FFTs which are each calculated by 2 butterflies followed by 2 multiplies. The third stage has 2 length-8 FFTs, each calculated by 4 butterflies followed by 8 multiplies and the last stage is simply 8 length-2 butterflies. This flow graph should be compared with the index map in Polynomial Description of Signals , the polynomial decomposition in The DFT as Convolution or Filtering , and the program in Appendix 3 . Here, the FFT must be preceded by a scrambler.

The third and fourth figures below are a length-16 decimation-in-frequency and a decimation-in-time but, in contrast to the figures above, the DIF has the output in order which requires a scrambled input and the DIT has the input in order which requires the output be unscrambled. Compare with the first two figures. Note the order of the twiddle factors. The number of additions and multiplications in all four flow graphs is the same and the structure of the three-loop program which executes the flow graph is the same.

This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from left to right in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, the third dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The third dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, the fifth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fifth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines connect to the eighth dots on the even numbered lines below them. There are various sections of the figure that are labeled with a variable. First, to the left of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots two and three for lines nine through sixteen, the horizontal lines are labeled in order w^0 through w^7 from top to bottom. In between dots four and five for lines five through eight are labels that read from top to bottom, w^0, w^2, w^4, w^6. In between dots four and five for lines thirteen through sixteen are labels that read from top to bottom, w^0, w^2, w^4, w^6. In between dots six and seven for lines three, seven, eleven, and fifteen is the label w^0. In between dots six and seven for lines four, eight, twelve, and sixteen is the label w^4. To the right of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15). This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from left to right in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, the third dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The third dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, the fifth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fifth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines connect to the eighth dots on the even numbered lines below them. There are various sections of the figure that are labeled with a variable. First, to the left of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots two and three for lines nine through sixteen, the horizontal lines are labeled in order w^0 through w^7 from top to bottom. In between dots four and five for lines five through eight are labels that read from top to bottom, w^0, w^2, w^4, w^6. In between dots four and five for lines thirteen through sixteen are labels that read from top to bottom, w^0, w^2, w^4, w^6. In between dots six and seven for lines three, seven, eleven, and fifteen is the label w^0. In between dots six and seven for lines four, eight, twelve, and sixteen is the label w^4. To the right of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15).
Length-16, Decimation-in-Frequency, In-order input, Radix-2 FFT
This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from right to left in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, the third dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The third dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, the fifth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fifth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines connect to the eighth dots on the even numbered lines below them. There are various sections of the figure that are labeled with a variable. First, to the right of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots two and three for lines nine through sixteen, the horizontal lines are labeled in order w^0 through w^7 from top to bottom. In between dots four and five for lines five through eight are labels that read from top to bottom, w^0, w^2, w^4, w^6. In between dots four and five for lines thirteen through sixteen are labels that read from top to bottom, w^0, w^2, w^4, w^6. In between dots six and seven for lines three, seven, eleven, and fifteen is the label w^0. In between dots six and seven for lines four, eight, twelve, and sixteen is the label w^4. To the left of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15). This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from right to left in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, the third dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The third dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, the fifth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fifth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines connect to the eighth dots on the even numbered lines below them. There are various sections of the figure that are labeled with a variable. First, to the right of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots two and three for lines nine through sixteen, the horizontal lines are labeled in order w^0 through w^7 from top to bottom. In between dots four and five for lines five through eight are labels that read from top to bottom, w^0, w^2, w^4, w^6. In between dots four and five for lines thirteen through sixteen are labels that read from top to bottom, w^0, w^2, w^4, w^6. In between dots six and seven for lines three, seven, eleven, and fifteen is the label w^0. In between dots six and seven for lines four, eight, twelve, and sixteen is the label w^4. To the left of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15).
Length-16, Decimation-in-Time, In-order output, Radix-2 FFT
This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from right to left in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, the third dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The third dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, the fifth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fifth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines connect to the eighth dots on the even numbered lines below them. There are various sections of the figure that are labeled with a variable. First, to the right of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots two and three on lines five through eight is the label W^0, and in between dots two and three on lines thirteen through sixteen is the label W^4. In between dots four and five on lines three and four is the label W^0. In between dots four and five on lines seven and eight is the label W^4. In between dots four and five on lines eleven and twelve is the label W^2. In between dots four and five on lines fifteen and sixteen is the label W^6. In between dots six and seven on even numbered lines are labeled from top to bottom W^0, W^4, W^2, W^6, W^1, W^5, W^3, W^7. To the left of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15). This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from right to left in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, the third dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The third dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, the fifth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fifth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines connect to the eighth dots on the even numbered lines below them. There are various sections of the figure that are labeled with a variable. First, to the right of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots two and three on lines five through eight is the label W^0, and in between dots two and three on lines thirteen through sixteen is the label W^4. In between dots four and five on lines three and four is the label W^0. In between dots four and five on lines seven and eight is the label W^4. In between dots four and five on lines eleven and twelve is the label W^2. In between dots four and five on lines fifteen and sixteen is the label W^6. In between dots six and seven on even numbered lines are labeled from top to bottom W^0, W^4, W^2, W^6, W^1, W^5, W^3, W^7. To the left of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15).
Length-16, alternate Decimation-in-Frequency, In-order output, Radix-2 FFT
This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from left to right in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, the third dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The third dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, the fifth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fifth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines connect to the eighth dots on the even numbered lines below them. There are various sections of the figure that are labeled with a variable. First, to the left of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots two and three on lines five through eight is the label W^0, and in between dots two and three on lines thirteen through sixteen is the label W^4. In between dots four and five on lines three and four is the label W^0. In between dots four and five on lines seven and eight is the label W^4. In between dots four and five on lines eleven and twelve is the label W^2. In between dots four and five on lines fifteen and sixteen is the label W^6. In between dots six and seven on even numbered lines are labeled from top to bottom W^0, W^4, W^2, W^6, W^1, W^5, W^3, W^7. To the right of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15). This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from left to right in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, the third dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The third dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, the fifth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fifth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines connect to the eighth dots on the even numbered lines below them. There are various sections of the figure that are labeled with a variable. First, to the left of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots two and three on lines five through eight is the label W^0, and in between dots two and three on lines thirteen through sixteen is the label W^4. In between dots four and five on lines three and four is the label W^0. In between dots four and five on lines seven and eight is the label W^4. In between dots four and five on lines eleven and twelve is the label W^2. In between dots four and five on lines fifteen and sixteen is the label W^6. In between dots six and seven on even numbered lines are labeled from top to bottom W^0, W^4, W^2, W^6, W^1, W^5, W^3, W^7. To the right of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15).
Length-16, alternate Decimation-in-Time, In-order input, Radix-2 FFT

The following is a length-16, decimation-in-frequency Radix-4 FFT with the input data in order and output data scrambled. There are two stages with the first stage having 4 length-4 "butterflies" followed by 12 multiplications by powers of W which are called "twiddle factors. The second stage has 4 length-4 FFTs which are each calculated by 4 butterflies followed by 4 multiplies. Note, each stage here looks like two stages but it is one and there is only one place where twiddle factor multiplications appear. This flow graph should be compared with the index map in Polynomial Description of Signals , the polynomial decomposition in The DFT as Convolution or Filtering , and the program in Appendix 3 . Log to the base 4 of 16 is 2. The total number of twiddle factor multiplication here is 12 compared to 24 for the radix-2. The unscrambler is a base-four reverse order counter rather than a bit reverse counter, however, a modification of the radix four butterflies will allow a bit reverse counter to be used with the radix-4 FFT as with the radix-2.

This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from left to right in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, the third dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The third dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, the fifth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fifth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines connect to the eighth dots on the even numbered lines below them. There are various sections of the figure that are labeled with a variable. First, to the left of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots two and three on lines thirteen through sixteen is the label -j. In between dots four and five for lines five through sixteen are labels that read from top to bottom, w^0, w^2, w^4, w^6, w^0, w^1, w^2, w^3, w^4, w^5, w^6, and w^7. In between dots six and seven for lines four, eight, twelve, sixteen is the label -j. To the right of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15). This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from left to right in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, the third dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The third dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, the fifth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fifth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines connect to the eighth dots on the even numbered lines below them. There are various sections of the figure that are labeled with a variable. First, to the left of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots two and three on lines thirteen through sixteen is the label -j. In between dots four and five for lines five through sixteen are labels that read from top to bottom, w^0, w^2, w^4, w^6, w^0, w^1, w^2, w^3, w^4, w^5, w^6, and w^7. In between dots six and seven for lines four, eight, twelve, sixteen is the label -j. To the right of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15).
Length-16, Decimation-in-Frequency, In-order input, Radix-4 FFT

The following two flowgraphs are length-16, decimation-in-frequency Split Radix FFTs with the input data in order and output data scrambled. Because the "butterflies" are L shaped, the stages do not progress uniformly like the Radix-2 or 4. These two figures are the same with the first drawn in a way to compare with the Radix-2 and 4, and the second to illustrate the L shaped butterflies. These flow graphs should be compared with the index map in Polynomial Description of Signals and the program in Appendix 3 . Because of the non-uniform stages, the program indexing is more complicated. Although the number of twiddle factor multiplications is 12 as was the radix-4 case, for longer lengths, the split-radix has slightly fewer multiplications than the radix-4.

Because the structures of the radix-2, radix-4, and split-radix FFTs are the same, the number of data additions is same for all of them. However, each complex twiddle factor multiplication requires two real additions (and four real multiplications) the number of additions will be fewer for the structures with fewer multiplications.

This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from left to right in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, the third dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The third dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, the fifth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fifth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines connect to the eighth dots on the even numbered lines below them. There are various sections of the figure that are labeled with a variable. First, to the left of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots two and three on lines thirteen through sixteen is the label -j. In between dots four and five for lines seven through sixteen are labels that read from top to bottom, -j, -j, w^0, w^1, w^2, w^3, w^4, w^5, w^6, and w^7. In between dots six and seven for lines four, twelve, sixteen is the label -j. In between dots six and seven for lines six through eight are labels that read from top to bottom, w^2, w^4, w^6. To the right of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15). This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from left to right in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, the third dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The third dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, the fifth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fifth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines connect to the eighth dots on the even numbered lines below them. There are various sections of the figure that are labeled with a variable. First, to the left of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots two and three on lines thirteen through sixteen is the label -j. In between dots four and five for lines seven through sixteen are labels that read from top to bottom, -j, -j, w^0, w^1, w^2, w^3, w^4, w^5, w^6, and w^7. In between dots six and seven for lines four, twelve, sixteen is the label -j. In between dots six and seven for lines six through eight are labels that read from top to bottom, w^2, w^4, w^6. To the right of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15).
Length-16, Decimation-in-Frequency, In-order input, Split-Radix FFT
This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from left to right in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, there is no third dot. The second dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The second dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, there is no fifth dot. The fourth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fourth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines two, six, eight, ten, and fourteen connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines one, five, seven, nine, and thirteen connect to the eighth dots on the even numbered lines below them. The sixth dots on even numbered lines four, twelve, and sixteen connect to the seventh dots on the odd numbered line above them. The sixth dots on odd numbered lines three, eleven, and fifteen connect to the seventh dots on the even numbered line below them. There are various sections of the figure that are labeled with a variable. First, to the left of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots four and five for lines nine through sixteen are labels that read from top to bottom, w^0, w^1, w^2, w^3, w^0, w^3, w^6, and w^9. In between dots six and seven for lines four, twelve, sixteen is the label -j. In between dots six and seven for lines five through eight are labels that read from top to bottom, w^0, w^2, w^4, w^6. To the right of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15). This is a complex figure consisting of sixteen horizontal lines connected by various diagonal lines. The diagonal lines connect to the horizontal lines at eight different points. The lines will be referenced by order from top to bottom, and the points will be referenced by numbered dots from left to right in this description. All of the connected diagonal segments referenced as a group move in the same direction systematically, do not cross paths, and connect only one dot to another. The first dot on lines one through eight are connected to the second dot on lines nine through sixteen, with each line crossing and moving down eight spaces. Conversely, the first dot on lines nine through sixteen connect to the second dot on lines one through eighteen. In the upper half of the next section, the third dots on lines one through four connect to the fourth dots on lines five through eight. Conversely, the third dots on lines five through eight connect to the fourth dots on lines one through four. In the lower half of the next section, there is no third dot. The second dots on lines nine through twelve connect to the fourth dots on lines thirteen through sixteen. The second dots on lines thirteen through sixteen also connect with the fourth dots on lines nine through twelve. The next section is visually divided into four parts. In part one, the fifth dots on lines one and two connect to the sixth dots on lines three and four, and the fifth dots on lines three and four connect to the sixth dots on lines one and two. In part two, there is no fifth dot. The fourth dots on lines five and six connect to the sixth dots on lines seven and eight, and the fourth dots on lines seven and eight connect to the sixth dots on lines five and six. In part three, the fifth dots on lines nine and ten connect to the sixth dots on lines eleven and twelve, and the fifth dots on lines eleven and twelve connect to the sixth dots on lines nine and ten. In part four, the fifth dots on lines thirteen and fourteen connect to the sixth dots on lines fifteen and sixteen, and the fifth dots on lines fifteen and sixteen connect to the sixth dots on lines thirteen and fourteen. In the final section of the figure, the diagonal lines are grouped into eight sections that all follow a similar pattern. The seventh dots on even numbered lines two, six, eight, ten, and fourteen connect to the eighth dots on the odd numbered line above them, and the seventh dots on odd numbered lines one, five, seven, nine, and thirteen connect to the eighth dots on the even numbered lines below them. The sixth dots on even numbered lines four, twelve, and sixteen connect to the seventh dots on the odd numbered line above them. The sixth dots on odd numbered lines three, eleven, and fifteen connect to the seventh dots on the even numbered line below them. There are various sections of the figure that are labeled with a variable. First, to the left of each horizontal lines and dot number one, the lines are labeled in order from x(0) to x(15) from top to bottom. In between dots four and five for lines nine through sixteen are labels that read from top to bottom, w^0, w^1, w^2, w^3, w^0, w^3, w^6, and w^9. In between dots six and seven for lines four, twelve, sixteen is the label -j. In between dots six and seven for lines five through eight are labels that read from top to bottom, w^0, w^2, w^4, w^6. To the right of the horizontal lines and the eighth dots are labels read from top to bottom, X(0), X(8), X(4), X(12), X(2), X(10), X(6), X(14), X(1), X(9), X(5), X(13), X(3), X(11), X(7), and X(15).
Length-16, Decimation-in-Frequency, Split-Radix with special BFs FFT

Questions & Answers

prostaglandin and fever
Maha Reply
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
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Source:  OpenStax, Test: no other way to test this than to publish it, sorry.. OpenStax CNX. May 18, 2009 Download for free at http://cnx.org/content/col10705/1.1
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