<< Chapter < Page Chapter >> Page >
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

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fast fourier transforms' conversation and receive update notifications?

Ask