EELE 4310: Digital Signal Processing (DSP) Chapter # 8 : Efficient - - PowerPoint PPT Presentation

EELE 4310: Digital Signal Processing (DSP)

Chapter # 8 : Efficient Computation of the DFT: Fast Fourier Transform Algorithms

Spring, 2012/2013

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

1 / 15

Outline

1

Efficient Computation of the DFT: FFT Algorithms

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

2 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 6

Remember from the last lecture that the N-DFT of x(n) can be represented as: X(k) = F1(k) + W k

NF2(k) ,

k = 0, 1, · · · , N

2 − 1

X(k + N

2 ) = F1(k) − W k NF2(k) ,

k = 0, 1, · · · , N

2 − 1

where F1(k) and F2(k) are the N/2-point DFTs of the sequences feven and fodd, respectively. Additionally, F1(k) can be represented as: F1(k) = V1(k) + W k

N/2V2(k) ,

k = 0, 1, · · · , N

4 − 1

F1(k + N

4 ) = V1(k) − W k N/2V2(k) ,

k = 0, 1, · · · , N

4 − 1

where V1(k) and V2(k) are the N/4-point DFTs of the sequences feven(2n) and feven(2n + 1), respectively. F2(k) also can be represented by the same way as F1(k). F2(k) = ψ1(k) + W k

N/2ψ2(k) ,

k = 0, 1, · · · , N

4 − 1

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

3 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 6

Remember from the last lecture that the N-DFT of x(n) can be represented as: X(k) = F1(k) + W k

NF2(k) ,

k = 0, 1, · · · , N

2 − 1

X(k + N

2 ) = F1(k) − W k NF2(k) ,

k = 0, 1, · · · , N

2 − 1

where F1(k) and F2(k) are the N/2-point DFTs of the sequences feven and fodd, respectively. Additionally, F1(k) can be represented as: F1(k) = V1(k) + W k

N/2V2(k) ,

k = 0, 1, · · · , N

4 − 1

F1(k + N

4 ) = V1(k) − W k N/2V2(k) ,

k = 0, 1, · · · , N

4 − 1

where V1(k) and V2(k) are the N/4-point DFTs of the sequences feven(2n) and feven(2n + 1), respectively. F2(k) also can be represented by the same way as F1(k). F2(k) = ψ1(k) + W k

N/2ψ2(k) ,

k = 0, 1, · · · , N

4 − 1

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

3 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 6

Remember from the last lecture that the N-DFT of x(n) can be represented as: X(k) = F1(k) + W k

NF2(k) ,

k = 0, 1, · · · , N

2 − 1

X(k + N

2 ) = F1(k) − W k NF2(k) ,

k = 0, 1, · · · , N

2 − 1

where F1(k) and F2(k) are the N/2-point DFTs of the sequences feven and fodd, respectively. Additionally, F1(k) can be represented as: F1(k) = V1(k) + W k

N/2V2(k) ,

k = 0, 1, · · · , N

4 − 1

F1(k + N

4 ) = V1(k) − W k N/2V2(k) ,

k = 0, 1, · · · , N

4 − 1

where V1(k) and V2(k) are the N/4-point DFTs of the sequences feven(2n) and feven(2n + 1), respectively. F2(k) also can be represented by the same way as F1(k). F2(k) = ψ1(k) + W k

N/2ψ2(k) ,

k = 0, 1, · · · , N

4 − 1

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

3 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 7

The basic computation performed at each stage is to:

Take tow complex numbers, say the pair (a, b). Multiply b by W r

N.

Add and subtract the product from a to form a new complex numbers (A, B).

The basic computation is called a butterfly because the flow graph resembles a butterfly.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

4 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 7

The basic computation performed at each stage is to:

Take tow complex numbers, say the pair (a, b). Multiply b by W r

N.

Add and subtract the product from a to form a new complex numbers (A, B).

The basic computation is called a butterfly because the flow graph resembles a butterfly.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

4 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 7

The basic computation performed at each stage is to:

Take tow complex numbers, say the pair (a, b). Multiply b by W r

N.

Add and subtract the product from a to form a new complex numbers (A, B).

The basic computation is called a butterfly because the flow graph resembles a butterfly.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

4 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 7

The basic computation performed at each stage is to:

Take tow complex numbers, say the pair (a, b). Multiply b by W r

N.

Add and subtract the product from a to form a new complex numbers (A, B).

The basic computation is called a butterfly because the flow graph resembles a butterfly.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

4 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 7

The basic computation performed at each stage is to:

Take tow complex numbers, say the pair (a, b). Multiply b by W r

N.

Add and subtract the product from a to form a new complex numbers (A, B).

The basic computation is called a butterfly because the flow graph resembles a butterfly.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

4 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 7

The basic computation performed at each stage is to:

Take tow complex numbers, say the pair (a, b). Multiply b by W r

N.

Add and subtract the product from a to form a new complex numbers (A, B).

The basic computation is called a butterfly because the flow graph resembles a butterfly.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

4 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 7

The basic computation performed at each stage is to:

Take tow complex numbers, say the pair (a, b). Multiply b by W r

N.

Add and subtract the product from a to form a new complex numbers (A, B).

The basic computation is called a butterfly because the flow graph resembles a butterfly.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

4 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 8

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

5 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 9

• Ex. Using the decimation in time FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

Check that DFT of x(n) = {2, 1, 2, 1} is X(k) = {6, 0, 2, 0}.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

6 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 10

Consider the case where N = 8, the first decimation yields the sequence x(0), x(2), x(4), x(6), x(1), x(3), x(5), x(7). The second decimation results in the sequence x(0), x(4), x(2), x(6), x(1), x(5), x(3), x(7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x(n) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order.

• Ex. the data point x(3) ≡ x(011) is placed in the position m = 110
• r m = 6 in the decimated array.

Therefore, the data x(n) after decimation is stored in bit-reversed

• rder.

If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

7 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 10

Consider the case where N = 8, the first decimation yields the sequence x(0), x(2), x(4), x(6), x(1), x(3), x(5), x(7). The second decimation results in the sequence x(0), x(4), x(2), x(6), x(1), x(5), x(3), x(7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x(n) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order.

• Ex. the data point x(3) ≡ x(011) is placed in the position m = 110
• r m = 6 in the decimated array.

Therefore, the data x(n) after decimation is stored in bit-reversed

• rder.

If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

7 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 10

Consider the case where N = 8, the first decimation yields the sequence x(0), x(2), x(4), x(6), x(1), x(3), x(5), x(7). The second decimation results in the sequence x(0), x(4), x(2), x(6), x(1), x(5), x(3), x(7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x(n) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order.

• Ex. the data point x(3) ≡ x(011) is placed in the position m = 110
• r m = 6 in the decimated array.

Therefore, the data x(n) after decimation is stored in bit-reversed

• rder.

If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

7 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 10

Consider the case where N = 8, the first decimation yields the sequence x(0), x(2), x(4), x(6), x(1), x(3), x(5), x(7). The second decimation results in the sequence x(0), x(4), x(2), x(6), x(1), x(5), x(3), x(7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x(n) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order.

• Ex. the data point x(3) ≡ x(011) is placed in the position m = 110
• r m = 6 in the decimated array.

Therefore, the data x(n) after decimation is stored in bit-reversed

• rder.

If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

7 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 10

Consider the case where N = 8, the first decimation yields the sequence x(0), x(2), x(4), x(6), x(1), x(3), x(5), x(7). The second decimation results in the sequence x(0), x(4), x(2), x(6), x(1), x(5), x(3), x(7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x(n) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order.

• Ex. the data point x(3) ≡ x(011) is placed in the position m = 110
• r m = 6 in the decimated array.

Therefore, the data x(n) after decimation is stored in bit-reversed

• rder.

If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

7 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 10

Consider the case where N = 8, the first decimation yields the sequence x(0), x(2), x(4), x(6), x(1), x(3), x(5), x(7). The second decimation results in the sequence x(0), x(4), x(2), x(6), x(1), x(5), x(3), x(7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x(n) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order.

• Ex. the data point x(3) ≡ x(011) is placed in the position m = 110
• r m = 6 in the decimated array.

Therefore, the data x(n) after decimation is stored in bit-reversed

• rder.

If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

7 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 10

Consider the case where N = 8, the first decimation yields the sequence x(0), x(2), x(4), x(6), x(1), x(3), x(5), x(7). The second decimation results in the sequence x(0), x(4), x(2), x(6), x(1), x(5), x(3), x(7). This shuffling of the input sequence has a well-defined order. By expressing the index n in the sequence x(n) in a binary form, we note that the order of the decimated data sequence is obtained by reading the binary representation of the index n in reversed order.

• Ex. the data point x(3) ≡ x(011) is placed in the position m = 110
• r m = 6 in the decimated array.

Therefore, the data x(n) after decimation is stored in bit-reversed

• rder.

If the input data is left in natural order, the output DFT sequence will occur in bit-reversed order.

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

7 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... 11

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

8 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is X(k) = N−1

n=0 x(n)W nk N ,

,k = 0, 1, · · · , N − 1

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is X(k) = N−1

n=0 x(n)W nk N ,

,k = 0, 1, · · · , N − 1 = (N/2)−1

n=0

x(n)W kn

N + N−1 n=N/2 x(n)W kn N

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is X(k) = N−1

n=0 x(n)W nk N ,

,k = 0, 1, · · · , N − 1 = (N/2)−1

n=0

x(n)W kn

N + N−1 n=N/2 x(n)W kn N

= (N/2)−1

n=0

x(n)W kn

N + W Nk/2 N

(N/2)−1

n=0

x(n + N

2 )W kn N

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is X(k) = N−1

n=0 x(n)W nk N ,

,k = 0, 1, · · · , N − 1 = (N/2)−1

n=0

x(n)W kn

N + N−1 n=N/2 x(n)W kn N

= (N/2)−1

n=0

x(n)W kn

N + W Nk/2 N

(N/2)−1

n=0

x(n + N

2 )W kn N

Since W Nk/2

N

= e−j2πNk/2N = e−jπk =

• e−jπk = (−1)k, then

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is X(k) = N−1

n=0 x(n)W nk N ,

,k = 0, 1, · · · , N − 1 = (N/2)−1

n=0

x(n)W kn

N + N−1 n=N/2 x(n)W kn N

= (N/2)−1

n=0

x(n)W kn

N + W Nk/2 N

(N/2)−1

n=0

x(n + N

2 )W kn N

Since W Nk/2

N

= e−j2πNk/2N = e−jπk =

• e−jπk = (−1)k, then

X(k) = (N/2)−1

n=0

• x(n) + (−1)kx(n + N

2 )

• W kn

N

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is X(k) = N−1

n=0 x(n)W nk N ,

,k = 0, 1, · · · , N − 1 = (N/2)−1

n=0

x(n)W kn

N + N−1 n=N/2 x(n)W kn N

= (N/2)−1

n=0

x(n)W kn

N + W Nk/2 N

(N/2)−1

n=0

x(n + N

2 )W kn N

Since W Nk/2

N

= e−j2πNk/2N = e−jπk =

• e−jπk = (−1)k, then

X(k) = (N/2)−1

n=0

• x(n) + (−1)kx(n + N

2 )

• W kn

N

By splitting (decimate) X(k) into even and odd numbered samples, we obtain,

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is X(k) = N−1

n=0 x(n)W nk N ,

,k = 0, 1, · · · , N − 1 = (N/2)−1

n=0

x(n)W kn

N + N−1 n=N/2 x(n)W kn N

= (N/2)−1

n=0

x(n)W kn

N + W Nk/2 N

(N/2)−1

n=0

x(n + N

2 )W kn N

Since W Nk/2

N

= e−j2πNk/2N = e−jπk =

• e−jπk = (−1)k, then

X(k) = (N/2)−1

n=0

• x(n) + (−1)kx(n + N

2 )

• W kn

N

By splitting (decimate) X(k) into even and odd numbered samples, we obtain, X(2k) = (N/2)−1

n=0

• x(n) + x(n + N

2 )

• W kn

N/2

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is X(k) = N−1

n=0 x(n)W nk N ,

,k = 0, 1, · · · , N − 1 = (N/2)−1

n=0

x(n)W kn

N + N−1 n=N/2 x(n)W kn N

= (N/2)−1

n=0

x(n)W kn

N + W Nk/2 N

(N/2)−1

n=0

x(n + N

2 )W kn N

Since W Nk/2

N

= e−j2πNk/2N = e−jπk =

• e−jπk = (−1)k, then

X(k) = (N/2)−1

n=0

• x(n) + (−1)kx(n + N

2 )

• W kn

N

By splitting (decimate) X(k) into even and odd numbered samples, we obtain, X(2k) = (N/2)−1

n=0

• x(n) + x(n + N

2 )

• W kn

N/2

= (N/2)−1

n=0

g1(n)W kn

N/2

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is X(k) = N−1

n=0 x(n)W nk N ,

,k = 0, 1, · · · , N − 1 = (N/2)−1

n=0

x(n)W kn

N + N−1 n=N/2 x(n)W kn N

= (N/2)−1

n=0

x(n)W kn

N + W Nk/2 N

(N/2)−1

n=0

x(n + N

2 )W kn N

Since W Nk/2

N

= e−j2πNk/2N = e−jπk =

• e−jπk = (−1)k, then

X(k) = (N/2)−1

n=0

• x(n) + (−1)kx(n + N

2 )

• W kn

N

By splitting (decimate) X(k) into even and odd numbered samples, we obtain, X(2k) = (N/2)−1

n=0

• x(n) + x(n + N

2 )

• W kn

N/2

= (N/2)−1

n=0

g1(n)W kn

N/2

and,

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is X(k) = N−1

n=0 x(n)W nk N ,

,k = 0, 1, · · · , N − 1 = (N/2)−1

n=0

x(n)W kn

N + N−1 n=N/2 x(n)W kn N

= (N/2)−1

n=0

x(n)W kn

N + W Nk/2 N

(N/2)−1

n=0

x(n + N

2 )W kn N

Since W Nk/2

N

= e−j2πNk/2N = e−jπk =

• e−jπk = (−1)k, then

X(k) = (N/2)−1

n=0

• x(n) + (−1)kx(n + N

2 )

• W kn

N

By splitting (decimate) X(k) into even and odd numbered samples, we obtain, X(2k) = (N/2)−1

n=0

• x(n) + x(n + N

2 )

• W kn

N/2

= (N/2)−1

n=0

g1(n)W kn

N/2

and, X(2k + 1) = (N/2)−1 x(n) − x(n + N )

• W n

W kn

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 1

The decimation in frequency FFT algorithm is obtained by using divide-and-conquer approach with choice of M = 2 and L = N/2. The N-point DFT of the sequence x(n) is X(k) = N−1

n=0 x(n)W nk N ,

,k = 0, 1, · · · , N − 1 = (N/2)−1

n=0

x(n)W kn

N + N−1 n=N/2 x(n)W kn N

= (N/2)−1

n=0

x(n)W kn

N + W Nk/2 N

(N/2)−1

n=0

x(n + N

2 )W kn N

Since W Nk/2

N

= e−j2πNk/2N = e−jπk =

• e−jπk = (−1)k, then

X(k) = (N/2)−1

n=0

• x(n) + (−1)kx(n + N

2 )

• W kn

N

By splitting (decimate) X(k) into even and odd numbered samples, we obtain, X(2k) = (N/2)−1

n=0

• x(n) + x(n + N

2 )

• W kn

N/2

= (N/2)−1

n=0

g1(n)W kn

N/2

and, X(2k + 1) = (N/2)−1 x(n) − x(n + N )

• W n

W kn

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

9 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 2

and, The basic computation of the decimation in frequency FFT algorithm involves the following butterfly operation

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

10 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 2

and, X(2k + 1) = (N/2)−1

n=0

• x(n) − x(n + N

2 )

• W n

N

• W kn

N/2

The basic computation of the decimation in frequency FFT algorithm involves the following butterfly operation

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

10 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 2

and, X(2k + 1) = (N/2)−1

n=0

• x(n) − x(n + N

2 )

• W n

N

• W kn

N/2

= (N/2)−1

n=0

g2(n)W kn

N/2

The basic computation of the decimation in frequency FFT algorithm involves the following butterfly operation

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

10 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 2

and, X(2k + 1) = (N/2)−1

n=0

• x(n) − x(n + N

2 )

• W n

N

• W kn

N/2

= (N/2)−1

n=0

g2(n)W kn

N/2

The basic computation of the decimation in frequency FFT algorithm involves the following butterfly operation

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

10 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 2

and, X(2k + 1) = (N/2)−1

n=0

• x(n) − x(n + N

2 )

• W n

N

• W kn

N/2

= (N/2)−1

n=0

g2(n)W kn

N/2

The basic computation of the decimation in frequency FFT algorithm involves the following butterfly operation

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

10 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 3

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

11 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 4

• Ex. Using the decimation in frequency FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

12 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 4

• Ex. Using the decimation in frequency FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

12 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 4

• Ex. Using the decimation in frequency FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

12 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 4

• Ex. Using the decimation in frequency FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

12 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 4

• Ex. Using the decimation in frequency FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

12 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 4

• Ex. Using the decimation in frequency FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

12 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 4

• Ex. Using the decimation in frequency FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

12 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 4

• Ex. Using the decimation in frequency FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

12 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 4

• Ex. Using the decimation in frequency FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

12 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... Decimation in Frequency ... 4

• Ex. Using the decimation in frequency FFT algorithm, compute the

4-point DFT of the sequence x(n) = {5, 0, −3, 4}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

12 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... IDFT Computation ... 1

The inverse discrete Fourier can be calculated using the same method but after changing the variable WN and multiplying the result by 1/N.

• Ex. Using the decimation in time IFFT algorithm, compute the

4-point IDFT of the sequence x(n) = {6, 8 + 4j, −2, 8 − 4j}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

13 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... IDFT Computation ... 1

The inverse discrete Fourier can be calculated using the same method but after changing the variable WN and multiplying the result by 1/N.

• Ex. Using the decimation in time IFFT algorithm, compute the

4-point IDFT of the sequence x(n) = {6, 8 + 4j, −2, 8 − 4j}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

13 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... IDFT Computation ... 2

• Ex. Using the decimation in frequency IFFT algorithm, compute the

4-point IDFT of the sequence x(n) = {6, 8 + 4j, −2, 8 − 4j}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

14 / 15

Efficient Computation of the DFT: FFT Algorithms

Radix-2 FFT Algorithms ... IDFT Computation ... 2

• Ex. Using the decimation in frequency IFFT algorithm, compute the

4-point IDFT of the sequence x(n) = {6, 8 + 4j, −2, 8 − 4j}

EELE 4310: Digital Signal Processing (DSP) - Ch.8

• Dr. Musbah Shaat

14 / 15

End of Chapter # 8