digital signal processing - kocaeli...
TRANSCRIPT
![Page 2: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/2.jpg)
Discrete Fourier Transform (DFT)
• DFT:
𝑋 𝑘 =
𝑛=0
𝑁−1
𝑥[𝑛]𝑒−𝑗(2𝜋/𝑁)𝑘𝑛 , 𝑘 = 0, 1, … , 𝑁 − 1
• IDFT:
𝑥 𝑛 =1
𝑁
𝑘=0
𝑁−1
𝑋[𝑘] 𝑒𝑗(2𝜋/𝑁)𝑘𝑛 , 𝑛 = 0, 1,… ,𝑁 − 1
![Page 3: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/3.jpg)
Windowing Prior to DFT
• The periodicity inherent in DFT may often cause some problems
• Consider the following sinusoidal signal. In the following slide, there is the signal that is obtained by the periodic repeating of this signal and the resulting DFT
![Page 4: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/4.jpg)
Windowing Prior to DFT
![Page 5: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/5.jpg)
Windowing Prior to DFT
• However, consider instead a sinusoidal signal that has not completed its full period:
• Repeating this signal periodically results in not a sinusoidal signal, which also affects its DFT:
![Page 6: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/6.jpg)
Windowing Prior to DFT
![Page 7: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/7.jpg)
Windowing Prior to DFT
• In this case, frequency components that are not actually in the signal occur
• This is caused by the potential discontinuity that is caused by the periodicity in DFT
• This can be considered as a leak of the energy to other frequencies, and therefore is termed spectral leakage
• This problem can be mitigated by windowing
![Page 8: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/8.jpg)
Windowing Prior to DFT
• Windowing is multiplying the signal with a function prior to DFT. The window function ensures that the signal’s amplitude converges to zero near the end boundaries, so that periodic repeating does not cause discontinuities
• Some common window functions are:
![Page 9: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/9.jpg)
Windowing Prior to DFT
• To multiply a signal with a window funtion, a window equal to the signal length is constructed. The effects are:
![Page 10: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/10.jpg)
Discrete Fourier Transform (DFT)
• DFT:
𝑋 𝑘 =
𝑛=0
𝑁−1
𝑥[𝑛]𝑒−𝑗(2𝜋/𝑁)𝑘𝑛 , 𝑘 = 0, 1, … , 𝑁 − 1
• IDFT:
𝑥 𝑛 =1
𝑁
𝑘=0
𝑁−1
𝑋[𝑘] 𝑒𝑗(2𝜋/𝑁)𝑘𝑛 , 𝑛 = 0, 1,… ,𝑁 − 1
![Page 11: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/11.jpg)
Discrete Fourier Transform (DFT)
• For each DFT coefficient, we have to compute 𝑁 complex multiplications and 𝑁 − 1 complex additions. In terms of real computations, this equates to 4𝑁 real multiplications and (4𝑁 − 2) real additions.
• For all DFT coefficients, N x N complex multiplications and 𝑁 ×(𝑁 − 1) complex additions. In terms of real computations, this equates to 4𝑁2 real multiplications and 𝑁 × (4𝑁 − 2) real additions.
• As N gets larger, the number of computations required for DFT becomes very large
![Page 12: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/12.jpg)
Fast Fourier Transform (FFT)
• The approaches that aim to provide a fast computation of DFT are termed fast Fourier transform (FFT). Most such approaches are motivated by the following two properties of DFT:
• 1) Periodicity:
𝑒−𝑗 2𝜋/𝑁 𝑘𝑛 = 𝑒−𝑗 2𝜋/𝑁 𝑘 𝑛+𝑁 = 𝑒−𝑗 2𝜋/𝑁 𝑘+𝑁 𝑛
• 2) Conjugate symmetry:
𝑒−𝑗 2𝜋/𝑁 𝑘 𝑁−𝑛 = 𝑒𝑗 2𝜋/𝑁 𝑘𝑛 = 𝑒−𝑗 2𝜋/𝑁 𝑘𝑛∗
![Page 13: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/13.jpg)
Fast Fourier Transform (FFT)
• There are two main FFT approaches:
• Decimation in time: x[n] is decomposed into successively smaller subsequences
• Decimation in frequency: X[k] is into successively smaller subsequences
• We will first go through Goertzel algorithm, which requires computation proportional to 𝑁2 but with a smaller constant of proportionality with respect to regular DFT. Later on we will examine more efficient FFT approaches.
![Page 14: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/14.jpg)
FFT: Goertzel Algorithm
• Let’s begin by noting that due to periodicity property, we have:
𝑒𝑗 2𝜋/𝑁 𝑘𝑁 = 𝑒𝑗2𝜋𝑘 = 1
• Using this property, we can write:
𝑋 𝑘 =
𝑛=0
𝑁−1
𝑥[𝑛]𝑒−𝑗(2𝜋/𝑁)𝑘𝑛 = 𝑒𝑗 2𝜋/𝑁 𝑘𝑁
𝑛=0
𝑁−1
𝑥[𝑛]𝑒−𝑗(2𝜋/𝑁)𝑘𝑛
=
𝑛=0
𝑁−1
𝑥[𝑛]𝑒𝑗(2𝜋/𝑁)𝑘 𝑁−𝑛
![Page 15: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/15.jpg)
FFT: Goertzel Algorithm
𝑋 𝑘 =
𝑟=0
𝑁−1
𝑥[𝑟]𝑒𝑗(2𝜋/𝑁)𝑘 𝑁−𝑟
• Since x[n] is zero for 𝑛 < 0 and 𝑛 ≥ 𝑁, we can state:
𝑋 𝑘 =
𝑟=0
𝑁−1
𝑥 𝑟 𝑒𝑗2𝜋𝑁 𝑘 𝑛−𝑟 𝑢[𝑛 − 𝑟] |𝑛=𝑁 = 𝑦𝑘[𝑛]|𝑛=𝑁
• 𝑦𝑘[𝑛] can be viewed as a discrete convolution of x[n] and the signal 𝑒𝑗 2𝜋 𝑁 𝑘𝑛𝑢[𝑛]
![Page 16: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/16.jpg)
FFT: Goertzel Algorithm
• Computing of each value of 𝑦𝑘[𝑛] requires 4 real multiplications and 4 real additions.
• To obtain X[k] for a particular value of k, we need to compute all the intervening values 𝑦𝑘 1 …𝑦𝑘[𝑁 − 1], which requires 4𝑁 real multiplications and 4𝑁 real additions.
• This would result in 4𝑁2 multiplications and 4𝑁2 additions for the complete X[k], for all k values.
• However, due to the system’s zero and pole locations (which we have not studied yet), this is reduced to 𝑁2 multiplications and 2𝑁2 additions.
![Page 17: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/17.jpg)
FFT: Decimation in Time
• FFT based on decimation in time depends on decomposing x[n] into successively smaller subsequences
• Consider N to be equal to 2𝑣. Since N is an even integer, we can consider computing X[k] by separating x[n] into two N/2 point sequences consisting of even and odd numbered points in x[n]:
𝑋 𝑘 =
𝑛 𝑒𝑣𝑒𝑛
𝑥[𝑛]𝑒−𝑗(2𝜋/𝑁)𝑘𝑛 +
𝑛 𝑜𝑑𝑑
𝑥[𝑛]𝑒−𝑗(2𝜋/𝑁)𝑘𝑛
![Page 18: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/18.jpg)
FFT: Decimation in Time
• Using 𝑛 = 2𝑟 for even 𝑛, and 𝑛 = 2𝑟 + 1 for odd 𝑛:
𝑋 𝑘 =
𝑟=0
𝑁/2−1
𝑥[2𝑟]𝑒−𝑗(2𝜋/𝑁)𝑘2𝑟 +
𝑟=0
𝑁/2−1
𝑥[2𝑟 + 1]𝑒−𝑗(2𝜋/𝑁)𝑘 2𝑟+1
=
𝑟=0
𝑁/2−1
𝑥[2𝑟]𝑊𝑁2𝑘𝑟 +𝑊𝑁
𝑘
𝑟=0
𝑁/2−1
𝑥[2𝑟 + 1]𝑊𝑁2𝑘𝑟
![Page 19: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/19.jpg)
FFT: Decimation in Time
𝑋 𝑘 =
𝑟=0
𝑁/2−1
𝑥[2𝑟]𝑊𝑁2𝑘𝑟 +𝑊𝑁
𝑘
𝑟=0
𝑁/2−1
𝑥[2𝑟 + 1]𝑊𝑁2𝑘𝑟
• However, 𝑊𝑁2 = 𝑒−2𝑗 2𝜋/𝑁 = 𝑒−𝑗 2𝜋/ 𝑁/2 = 𝑊𝑁/2. Thefore:
𝑋 𝑘 =
𝑟=0
𝑁/2−1
𝑥[2𝑟]𝑊𝑁/2𝑘𝑟 +𝑊𝑁
𝑘
𝑟=0
𝑁/2−1
𝑥[2𝑟 + 1]𝑊𝑁/2𝑘𝑟
= 𝐺 𝑘 +𝑊𝑁𝑘𝐻 𝑘 , 𝑘 = 0 , 1 , … , N − 1
where 𝐺 𝑘 and 𝐻 𝑘 are N/2 point DFTs of even and odd x[n] samples, and are periodic with N/2
![Page 20: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/20.jpg)
FFT: Decimation in Time
𝑋 𝑘 = 𝐺 𝑘 +𝑊𝑁𝑘𝐻 𝑘 , 𝑘 = 0 , 1 , … , N − 1
Using the property 𝑊𝑁𝑘+𝑁/2
= −𝑊𝑁𝑘, we can write:
𝑋 𝑘 = 𝐺 𝑘 +𝑊𝑁𝑘𝐻 𝑘 , 𝑘 = 0 , 1 , … , N/2 − 1
𝑋 𝑘 + N/2 = 𝐺 𝑘 −𝑊𝑁𝑘𝐻 𝑘 , 𝑘 = 0 , 1 , … , N/2 − 1
Note that 𝐺 𝑘 = 𝐺 𝑘 + 𝑁/2 and 𝐻 𝑘 = 𝐻 𝑘 + 𝑁/2
![Page 21: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/21.jpg)
FFT: Decimation in Time
![Page 22: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/22.jpg)
FFT: Decimation in Time
• This approach involves two 𝑁/2-point DFTs, which requires 2 𝑁/2 2 complex multiplications and approximately 2 𝑁/2 2
complex additions
• Combining these two 𝑁/2-point DFTs in turn requires 𝑁 complex additions. Therefore the total computation for this approach is N + 2 𝑁/2 2 complex multiplications and additions
• Note that (𝑁 + 𝑁2/2) < 𝑁2 when 𝑁 > 2
• This was the case when we broke the DFT computation into 𝑁/2-point DFTs. But we can continue this line of thought to obtain:
![Page 23: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/23.jpg)
FFT: Decimation in Time
• If 𝑁 = 2𝑣, this decimation in time can be done 𝑣 = log2𝑁times
• After a decimation of 𝑣 = log2𝑁 times, the number of complex multiplications and additions is equal to:
𝑁𝑣 = 𝑁 log2𝑁
![Page 24: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/24.jpg)
FFT: Decimation in Time
![Page 25: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/25.jpg)
FFT: Decimation in Time
![Page 26: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/26.jpg)
FFT: Decimation in Time
• Note that𝑊𝑁0 = 𝑒−𝑗 2𝜋/𝑁 0 = 1
𝑊𝑁𝑁/2= 𝑒−𝑗 2𝜋/𝑁 𝑁/2 = 𝑒−𝑗𝜋 = −1
• So the left-most stage simplifies to:
![Page 27: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/27.jpg)
FFT: Decimation in Time
• Computing for a three digit x[n], we have:
111x111X7x7X
011x110X3x6X
101x101X5x5X
001x100X1x4X
110x011X6x3X
010x010X2x2X
100x001X4x1X
000x000X0x0X
00
00
00
00
00
00
00
00
![Page 28: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/28.jpg)
FFT: Decimation in Time
• Example: 𝑥 𝑛 = 1 3 0 2 4 1 0 2 ⇒ FFT time-decimation:
𝑥𝑜𝑠 𝑛 = 1 0 4 0 , 𝑥𝑒𝑠 𝑛 = 3 2 1 2
𝑥𝑜𝑜𝑠 𝑛 = 1 4 , 𝑥𝑜𝑒𝑠 𝑛 = 0 0 , 𝑥𝑒𝑜𝑠 𝑛 = 3 1 , 𝑥𝑒𝑒𝑠 𝑛 = 2 2
𝑋𝑜𝑜𝑠 𝑘 = 5 − 3 , 𝑋𝑜𝑒𝑠 𝑘 = 0 0 , 𝑋𝑒𝑜𝑠 𝑘 = 4 2 , 𝑋𝑒𝑒𝑠 𝑘 = 4 0
𝑋𝑜𝑠 𝑘 = 𝑋𝑜𝑜𝑠 𝑘 +𝑊4𝑘𝑋𝑜𝑒𝑠 𝑘
![Page 29: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/29.jpg)
FFT: Decimation in Time
• Example (continued):
𝑋𝑜𝑜𝑠 𝑘 = 5 − 3 , 𝑋𝑜𝑒𝑠 𝑘 = 0 0 , 𝑋𝑒𝑜𝑠 𝑘 = 4 2 , 𝑋𝑒𝑒𝑠 𝑘 = 4 0
𝑋𝑜𝑠 𝑘 = 𝑋𝑜𝑜𝑠 𝑘 +𝑊4𝑘𝑋𝑜𝑒𝑠 𝑘 , 𝑘 = 0,1
𝑋𝑜𝑠 𝑘 + 2 = 𝑋𝑜𝑜𝑠 𝑘 −𝑊4𝑘𝑋𝑜𝑒𝑠 𝑘 , 𝑘 = 0,1
𝑋𝑜𝑠 0 = 5 + 𝑒−𝑗 2𝜋/4 0 0 = 5
𝑋𝑜𝑠 1 = −3 + 𝑒−𝑗2𝜋4 1 0 = −3
𝑋𝑜𝑠 2 = 5 − 𝑒−𝑗 2𝜋/4 0 0 = 5
𝑋𝑜𝑠 3 = −3 − 𝑒−𝑗2𝜋4 1 0 = −3
𝑋𝑜𝑠 𝑘 = [5 − 3 5 − 3]
![Page 30: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/30.jpg)
FFT: Decimation in Time
• Example (continued):
𝑋𝑜𝑜𝑠 𝑘 = 5 − 3 , 𝑋𝑜𝑒𝑠 𝑘 = 0 0 , 𝑋𝑒𝑜𝑠 𝑘 = 4 2 , 𝑋𝑒𝑒𝑠 𝑘 = 4 0
𝑋𝑒𝑠 𝑘 = 𝑋𝑒𝑜𝑠 𝑘 +𝑊4𝑘𝑋𝑒𝑒𝑠 𝑘 , 𝑘 = 0,1
𝑋𝑒𝑠 𝑘 + 2 = 𝑋𝑒𝑜𝑠 𝑘 −𝑊4𝑘𝑋𝑒𝑒𝑠 𝑘 , 𝑘 = 0,1
𝑋𝑒𝑠 0 = 4 + 𝑒−𝑗 2𝜋/4 0 4 = 8
𝑋𝑒𝑠 1 = 2 + 𝑒−𝑗2𝜋4 1 0 = 2
𝑋𝑒𝑠 2 = 4 − 𝑒−𝑗 2𝜋/4 0 4 = 0
𝑋𝑒𝑠 3 = 2 − 𝑒−𝑗2𝜋4 1 0 = 2
𝑋𝑒𝑠 𝑘 = [8 2 0 2]
![Page 31: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/31.jpg)
FFT: Decimation in Time
• Example (continued):
𝑋𝑜𝑠 𝑘 = 5 − 3 5 − 3 , 𝑋𝑒𝑠 𝑘 = [8 2 0 2]
𝑋 𝑘 = 𝑋𝑜𝑠 𝑘 +𝑊8𝑘𝑋𝑒𝑠 𝑘 , 𝑘 = 0,1,2,3
𝑋 𝑘 + 4 = 𝑋𝑜𝑠 𝑘 −𝑊8𝑘𝑋𝑒𝑠 𝑘 , 𝑘 = 0,1,2,3
𝑋 0 = 5 + 𝑒−𝑗 2𝜋/8 0 8 = 13
𝑋 1 = −3 + 𝑒−𝑗2𝜋8 1 2 = −1.5858 − 𝑗1.412
𝑋 2 = 5 + 𝑒−𝑗 2𝜋/8 2 0 = 5
𝑋 3 = −3 + 𝑒−𝑗2𝜋8 3 2 = −4.4142 − 𝑗1.4142
![Page 32: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/32.jpg)
FFT: Decimation in Time
• Example (continued):
𝑋𝑜𝑠 𝑘 = 5 − 3 5 − 3 , 𝑋𝑒𝑠 𝑘 = [8 2 0 2]
𝑋 𝑘 = 𝑋𝑜𝑠 𝑘 +𝑊8𝑘𝑋𝑒𝑠 𝑘 , 𝑘 = 0,1,2,3
𝑋 𝑘 + 4 = 𝑋𝑜𝑠 𝑘 −𝑊8𝑘𝑋𝑒𝑠 𝑘 , 𝑘 = 0,1,2,3
𝑋 4 = 5 − 𝑒−𝑗2𝜋8 0 8 = −3
𝑋 5 = −3 − 𝑒−𝑗2𝜋8 1 2 = −4.4142 + 𝑗1.4142
𝑋 6 = 5 − 𝑒−𝑗 2𝜋/8 2 0 = 5
𝑋 7 = −3 − 𝑒−𝑗2𝜋8 3 2 = −1.5858 + 𝑗1.412
![Page 33: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/33.jpg)
FFT: Decimation in Time
• Example (continued):
𝑥 𝑛 = 1 3 0 2 4 1 0 2
𝑋 0 = 13𝑋 1 = −1.5858 − 𝑗1.412
𝑋 2 = 5𝑋 3 = −4.4142 − 𝑗1.4142
𝑋 4 = −3𝑋 5 = −4.4142 + 𝑗1.4142
𝑋 6 = 5𝑋 7 = −1.5858 + 𝑗1.412
![Page 34: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/34.jpg)
FFT: Decimation in Frequency
• FFT based on decimation in frequency depends on decomposing X[k] into successively smaller subsequences
• Consider N to be equal to 2𝑣. Since N is an even integer, we can consider computing even numbered frequency samples and odd numbered frequency samples separately
• Standard DFT equation is:
𝑋 𝑘 =
𝑛=0
𝑁−1
𝑥[𝑛]𝑒−𝑗(2𝜋/𝑁)𝑘𝑛 =
𝑛=0
𝑁−1
𝑥[𝑛]𝑊𝑁𝑛𝑘
![Page 35: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/35.jpg)
FFT: Decimation in Frequency
• The even samples of X[k] are:
𝑋 2𝑟 =
𝑛=0
𝑁−1
𝑥[𝑛]𝑊𝑁𝑛(2𝑟)
=
𝑛=0
𝑁/2−1
𝑥[𝑛]𝑊𝑁𝑛(2𝑟)+
𝑛=𝑁/2
𝑁−1
𝑥[𝑛]𝑊𝑁𝑛(2𝑟)
=
𝑛=0
𝑁/2−1
𝑥[𝑛]𝑊𝑁𝑛(2𝑟)+
𝑛=0
𝑁/2−1
𝑥[𝑛 + 𝑁 2]𝑊𝑁𝑛+𝑁/2 (2𝑟)
![Page 36: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/36.jpg)
FFT: Decimation in Frequency
𝑋 2𝑟 =
𝑛=0
𝑁/2−1
𝑥[𝑛]𝑊𝑁𝑛(2𝑟)+
𝑛=0
𝑁/2−1
𝑥[𝑛 + 𝑁 2]𝑊𝑁𝑛+𝑁/2 (2𝑟)
• Due to periodicity, 𝑊𝑁𝑛+𝑁/2 (2𝑟)
= 𝑊𝑁2𝑟𝑛𝑊𝑁
𝑟𝑁 = 𝑊𝑁2𝑟𝑛
𝑋 2𝑟 =
𝑛=0
𝑁/2−1
𝑥 𝑛 + 𝑥[𝑛 + 𝑁 2] 𝑊𝑁/2𝑛𝑟
![Page 37: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/37.jpg)
FFT: Decimation in Frequency
𝑋 2𝑟 =
𝑛=0
𝑁/2−1
𝑥 𝑛 + 𝑥[𝑛 + 𝑁 2] 𝑊𝑁/2𝑛𝑟
• Using a similar approach, we obtain the odd samples as:
𝑋 2𝑟 + 1 =
𝑛=0
𝑁/2−1
𝑥 𝑛 − 𝑥[𝑛 + 𝑁 2] 𝑊𝑁𝑛𝑊𝑁/2𝑛𝑟
![Page 38: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/38.jpg)
FFT: Decimation in Frequency
![Page 39: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/39.jpg)
FFT: Decimation in Frequency
![Page 40: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/40.jpg)
FFT: Decimation in Frequency
![Page 41: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/41.jpg)
FFT: Decimation in Frequency
• Example: 𝑥 𝑛 = 1 3 0 2 4 1 0 2 ⇒ FFT freq.-decimation:
• N=8 DFT:
𝑋1 𝑘 = 𝑋 2𝑟 =
𝑛=0
𝑁/2−1
𝑥 𝑛 + 𝑥[𝑛 + 𝑁 2] 𝑊𝑁/2𝑛𝑟
=
𝑛=0
3
𝑥 𝑛 + 𝑥[𝑛 + 4] 𝑊4𝑛𝑟 , 𝑘 = 0, 1, 2, 3
𝑋2 𝑘 = 𝑋 2𝑟 + 1 =
𝑛=0
𝑁/2−1
𝑥 𝑛 − 𝑥[𝑛 + 𝑁 2] 𝑊𝑁𝑛𝑊𝑁/2𝑛𝑟
=
𝑛=0
3
𝑥 𝑛 − 𝑥[𝑛 + 4] 𝑊8𝑛𝑊4𝑛𝑟 , 𝑘 = 0, 1, 2, 3
![Page 42: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/42.jpg)
FFT: Decimation in Frequency
• Example (continued): 𝑥 𝑛 = 1 3 0 2 4 1 0 2
𝑥1[𝑛] = 𝑥 𝑛 + 𝑥 𝑛 + 4 = [5 4 0 4]
𝑥2[𝑛] = 𝑥 𝑛 − 𝑥 𝑛 + 4 𝑊8𝑛
= −3 2 0 0 𝑊8𝑛
= [−3𝑒−𝑗2𝜋8 0 2𝑒
−𝑗2𝜋8 1 0𝑒
−𝑗2𝜋8 2 0𝑒−𝑗 2𝜋/8 3]
= [−3 1.4142 − 𝑗1.4142 0 0]
![Page 43: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/43.jpg)
FFT: Decimation in Frequency
• Example (continued): 𝑥1[𝑛] = [5 4 0 4]
• N=4 DFT:
𝑋11 𝑘 = 𝑋1 2𝑘 =
𝑛=0
1
𝑥1 𝑛 + 𝑥1 𝑛 + 2 𝑊2𝑘𝑛 , 𝑘 = 0, 1
𝑋12 𝑘 = 𝑋1 2𝑘 + 1 =
𝑛=0
1
𝑥1 𝑛 − 𝑥1 𝑛 + 2 𝑊4𝑛𝑊2𝑘𝑛 , 𝑘 = 0, 1
𝑥11 𝑛 = 𝑥1 𝑛 + 𝑥1 𝑛 + 2 = 5 8
𝑥12 𝑛 = 𝑥1 𝑛 − 𝑥1 𝑛 + 2 𝑊4𝑛 = 5 0
![Page 44: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/44.jpg)
FFT: Decimation in Frequency
Example (continued): 𝑥2[𝑛] = [−3 1.4142 − 𝑗1.4142 0 0]
• N=4 DFT:
𝑋21 𝑘 = 𝑋2 2𝑘 =
𝑛=0
1
𝑥2 𝑛 + 𝑥2 𝑛 + 2 𝑊2𝑘𝑛 , 𝑘 = 0, 1
𝑋22 𝑘 = 𝑋2 2𝑘 + 1 =
𝑛=0
1
𝑥2 𝑛 − 𝑥2 𝑛 + 2 𝑊4𝑛𝑊2𝑘𝑛 , 𝑘 = 0, 1
𝑥21 𝑛 = 𝑥2 𝑛 + 𝑥2 𝑛 + 2 = [−3 1.4142 − 𝑗1.4142 ]
𝑥22 𝑛 = 𝑥2 𝑛 − 𝑥2 𝑛 + 2 𝑊4𝑛 = [−3 1.4142 − 𝑗1.4142 𝑒−𝑗2𝜋/4]
= [−3 −1.4142 − 𝑗1.4142 ]
![Page 45: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/45.jpg)
FFT: Decimation in Frequency
Example (continued):𝑥11 𝑛 = 5 8 ⇒ 𝑋11 𝑘 = [13 − 3]
𝑥12 𝑛 = 5 0 ⇒ 𝑋12 𝑘 = [5 5]
𝑥21 𝑛 = −3 1.4142 − 𝑗1.4142
⇒ 𝑋21 𝑘 = −1.5858 − 𝑗1.4142 −4.4142 − 𝑗1.4142
𝑥22 𝑛 = [−3 −1.4142 − 𝑗1.4142 ]
⇒ 𝑋22 𝑘 = [ −4.4142 − 𝑗1.4142 −1.5858 + 𝑗1.4142 ]
![Page 46: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/46.jpg)
FFT: Decimation in Frequency
Example (continued):𝑋11 𝑘 = 13 − 3 , 𝑋12 𝑘 = [5 5]
⇒ 𝑋1 𝑘 = 13 5 − 3 5
𝑋21 𝑘 = −1.5858 − 𝑗1.4142 −4.4142 − 𝑗1.4142
𝑋22 𝑘 = [ −4.4142 − 𝑗1.4142 −1.5858 + 𝑗1.4142 ]
⇒ 𝑋2 𝑘= −1.5858 − 𝑗1.4142 (−4.4142 − 𝑗1.4142 (−4.4142
![Page 47: Digital Signal Processing - Kocaeli Üniversitesiehm.kocaeli.edu.tr/dersnotlari_data/aerturk/Digital...Fast Fourier Transform (FFT) •The approaches that aim to provide a fast computation](https://reader031.vdocuments.mx/reader031/viewer/2022021819/5acebbe87f8b9a71028bcc28/html5/thumbnails/47.jpg)
FFT: Decimation in Frequency
Example (continued):
𝑋1 𝑘 = 13 5 − 3 5
𝑋2 𝑘 = −1.5858 − 𝑗1.4142 (−4.4142 − 𝑗1.4142−4.4142𝑗1.4142 −1.5858 + 𝑗1.4142 ]
⇒ 𝑋 𝑘= [13 −1.5858 − 𝑗1.4142 5 −4.4142 − 𝑗1.4142−3 −4.4142𝑗1.4142 5 −1.5858 + 𝑗1.4142 ]