1. Home
  2. Test: no other way to test this
  3. Test: no other way to test this
  4. Appendix 1: fft flowgraphs

0.3 Appendix 1: fft flowgraphs

<< Chapter < Page
  Test: no other way to test this   Page 1 / 1
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

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Test: no other way to test this than to publish it, sorry.' conversation and receive update notifications?

Ask