DFT Frequency (9A) Each ow of the DFT Matrix
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9A DFT Frequency 3
=8 DFT : The 1st ow of the DFT Matrix e j 4 0 e j 4 0 e j 4 0 e j 4 0 e j 4 0 e j 4 0 e j 4 0 e j 4 0 0 cycle W 8 k n = e j 2 8 k n k = 0, n = 0, 1,..., 7 cos t = cost t = 2 f t sin t = sint 2 0 8 t X[0] measures how much of the above signal component is present in x. T = Sampling Time Sampling Frequency = 1 Sequence Time Length T = Zero Frequency 9A DFT Frequency 4
=8 DFT : The 2nd ow of the DFT Matrix e j 4 0 e j 4 1 e j 4 2 e j 4 3 e j 4 4 e j 4 5 e j 4 6 e j 4 7 1 cycle W 8 k n = e j 2 8 k n k = 1, n = 0, 1,..., 7 cos t = cost t = 2 f t sin t = sint 2 1 8 t X[1] measures how much of the above signal component is present in x. T = Sampling Time Sampling Frequency Sequence Time Length T = 1 st Harmonic Freq = 1 f 1 = 1 T = 1 = 9A DFT Frequency 5
=8 DFT : The 3rd ow of the DFT Matrix e j 4 0 e j 4 2 e j 4 4 e j 4 6 e j 4 0 e j 4 2 e j 4 4 e j 4 6 2 cycles W 8 k n = e j 2 8 k n k = 2, n = 0, 1,..., 7 cos t = cost t = 2 f t sin t = sint 2 2 8 t X[2] measures how much of the above signal component is present in x. T = Sampling Time Sampling Frequency Sequence Time Length T = 2 nd Harmonic Freq = 1 f 2 = 2 T = 2 = 2 9A DFT Frequency 6
=8 DFT : The 4th ow of the DFT Matrix e j 4 0 e j 4 3 e j 4 6 e j 4 1 e j 4 4 e j 4 7 e j 4 2 e j 4 5 3 cycles W 8 k n = e j 2 8 k n k = 3, n = 0, 1,..., 7 cos t = cost t = 2 f t sin t = sint 2 3 8 t X[3] measures how much of the above signal component is present in x. T = Sampling Time Sampling Frequency Sequence Time Length T = 3 rd Harmonic Freq = 1 f 3 = 3 T = 3 = 3 9A DFT Frequency 7
=8 DFT : The 5th ow of the DFT Matrix e j 4 0 e j 4 4 e j 4 0 e j 4 4 e j 4 0 e j 4 4 e j 4 0 e j 4 4 4 cycles W 8 k n = e j 2 8 k n k = 4, n = 0, 1,..., 7 cos t = cost t = 2 f t sin t = sint 2 4 8 t X[4] measures how much of the above signal component is present in x. T = Sampling Time Sampling Frequency Sequence Time Length T = 4 th Harmonic Freq = 1 f 4 = 4 T = 4 = 4 9A DFT Frequency 8
=8 DFT : The 6th ow of the DFT Matrix e j 4 0 e j 4 5 e j 4 2 e j 4 7 e j 4 4 e j 4 1 e j 4 6 e j 4 3 3 cycles W 8 k n = e j 2 8 k n k = 5, n = 0, 1,..., 7 cos' t = cos t t = 2 f t sin' t = sin t 2 3 8 t X[5] measures how much of the above signal component is present in x. T = Sampling Time Sampling Frequency Sequence Time Length T = 3 rd Harmonic Freq = 1 f 3 = 3 T = 3 = 3 9A DFT Frequency 9
=8 DFT : The 7th ow of the DFT Matrix e j 4 0 e j 4 6 e j 4 4 e j 4 2 e j 4 0 e j 4 6 e j 4 4 e j 4 2 2 cycles W 8 k n = e j 2 8 k n k = 2, n = 0, 1,..., 7 cos' t = cos t t = 2 f t sin' t = sin t 2 2 8 t X[6] measures how much of the above signal component is present in x. T = Sampling Time Sampling Frequency Sequence Time Length T = 2 nd Harmonic Freq = 1 f 2 = 2 T = 2 = 2 9A DFT Frequency 10
=8 DFT : The 8th ow of the DFT Matrix e j 4 0 e j 4 7 e j 4 6 e j 4 5 e j 4 4 e j 4 3 e j 4 2 e j 4 1 1 cycle W 8 k n = e j 2 8 k n k = 7, n = 0, 1,..., 7 cos' t = cos t t = 2 f t sin' t = sin t 2 1 8 t X[7] measures how much of the above signal component is present in x. T = Sampling Time Sampling Frequency Sequence Time Length T = 1 st Harmonic Freq = 1 f 1 = 1 T = 1 = 9A DFT Frequency 11
Fundamental Frequency e j 4 0 e j 4 1 e j 4 2 e j 4 3 e j 4 4 e j 4 5 e j 4 6 e j 4 7 1 cycle T = Sampling Time Sampling Frequency Sequence Time Length T = 1 st Harmonic Freq = 1 f 1 = 1 T = 1 = f 2 = 2 f 1 f 3 = 3 f 1... f 1 = 1 f 1 Fundamental Frequency f o The Lowest Frequency in a harmonic series. f 0 = f 1 = 9A DFT Frequency 12
ormalized Frequency e j 4 0 e j 4 1 e j 4 2 e j 4 3 e j 4 4 e j 4 5 e j 4 6 e j 4 7 1 cycle T = Sampling Time Sampling Frequency = 1 (samples per second) Sequence Time Length f 1 = 1 f 1 1/ f 2 = 2 f 1 2/ f 3 = 3 f 1 3/... f 1 = 1 f 1 T = 1/ 1 st Harmonic Freq n = 0, 1, 2,..., 1 f 1 = 1 T = 1 = ormalized Frequency f n = n (cycles per sample) f n = n 9A DFT Frequency 13
egative Frequency (1) Coordinate (B) Coordinate (A) Coordinate (B) redraw = t = ' t ' = sin' t = sin t 9A DFT Frequency 14
egative Frequency (2) Coordinate (B) Coordinate (A) Coordinate (B) redraw = t = ' t ' = cos' t = cos t 9A DFT Frequency 15
Euler Equation (1) e j t = cost j sin t x real Top View y imag x real z time y imag Front View z time z time phase y imag Side View 2 x real time 9A DFT Frequency 16
egative Frequency (3) y x z y x z 9A DFT Frequency 17
egative Frequency 2 cycles W 8 k n = e j 2 8 k n k = 2, n = 0, 1,..., 7 cos t = cost t = 2 f t sin t = sint 2 2 8 t X[2] measures how much of the above signal component is present in x. T = Sampling Time Sampling Frequency Sequence Time Length T = 2 nd Harmonic Freq = 1 f 2 = 2 T = 2 = 2 9A DFT Frequency 18
=8 DFT : DFT Matrix in + or Frequencies 0 = 2 0th row: samples of cos 0 0 t j sin 0 0 t 1th row: samples of cos 1 0 t j sin 1 0 t 2th row: samples of cos 2 0 t j sin 2 0 t 3th row: samples of cos 3 0 t j sin 3 0 t 4th row: samples of cos 4 0 t j sin 4 0 t 5th row: samples of cos 5 0 t j sin 5 0 t 6th row: samples of cos 6 0 t j sin 6 0 t 7th row: samples of cos 7 0 t j sin 7 0 t (0 cycle) (1 cycle) (2 cycles) (3 cycles) (4 cycles) (5 cycles) (6 cycles) (7 cycles) 0th row: samples of cos 0 0 t j sin 0 0 t 1th row: samples of cos 7 0 t j sin7 0 t 2th row: samples of cos 6 0 t j sin6 0 t 3th row: samples of cos 5 0 t j sin 5 0 t 4th row: samples of cos 4 0 t j sin4 0 t 5th row: samples of cos3 0 t j sin3 0 t 6th row: samples of cos 2 0 t j sin 2 0 t 7th row: samples of cos1 0 t j sin1 0 t (0 cycle) (7 cycles) (6 cycles) (5 cycles) (4 cycles) (3 cycles) (2 cycles) (1 cycles) 9A DFT Frequency 19
=8 DFT : DFT Matrix in Both Frequencies 0 = 2 0th row: samples of cos 0 0 t j sin 0 0 t 1th row: samples of cos 1 0 t j sin 1 0 t 2th row: samples of cos 2 0 t j sin 2 0 t 3th row: samples of cos 3 0 t j sin 3 0 t 4th row: samples of cos 4 0 t j sin 4 0 t 5th row: samples of cos 5 0 t j sin 5 0 t 6th row: samples of cos 6 0 t j sin 6 0 t 7th row: samples of cos 7 0 t j sin 7 0 t (0 cycle) (1 cycle) (2 cycles) (3 cycles) (4 cycles) (5 cycles) (6 cycles) (7 cycles) 0th row: samples of cos 0 0 t j sin 0 0 t 1th row: samples of cos 1 0 t j sin 1 0 t 2th row: samples of cos 2 0 t j sin 2 0 t 3th row: samples of cos 3 0 t j sin 3 0 t 4th row: samples of cos 4 0 t j sin 4 0 t 5th row: samples of cos 3 0 t j sin3 0 t 6th row: samples of cos 2 0 t j sin 2 0 t 7th row: samples of cos 1 0 t j sin1 0 t (0 cycle) (1 cycle) (2 cycles) (3 cycles) (4 cycles) (3 cycles) (2 cycles) (1 cycles) 9A DFT Frequency 20
Frequency View of a DFT Matrix row 0 row 1 row 2 f = 0 f = 1 f o f = 2 f o 0cycle 1cycle 2cycles 0 1 2 ormalized Frequency 2 1 f o = row 2 1 row 2 row 2 1 f = 2 1 f o f = 2 f o f = 2 1 f o 2 2 2 1 cycles cycles 1 cycles 1 2 1 1 2 1 2 1 2 1 row 2 row 1 f = 2 f o f = 1 f o 2cycles 1cycle 2 1 9A DFT Frequency 21
Frequency View of a X[i] Vector X [0] f = 0 0 0 0cycle 2 1 X [1] X [2] f = 1 f o f = 2 f o 1 2 ormalized Frequency 1 2 1cycle 2cycles X [ 1] 2 X [ ] 2 f = 1 f 2 o f = f 2 o 1 2 f s 1 f 2 s 1 2 1 1 2 2 2 1 cycles cycles X [ 2 1] f = 2 1 f o 2 1 1 2 1 2 1 cycles 2 1 X [ 2] X [ 1] f = 2 f o f = 1 f o 2 1 2 1 2cycles 1cycle 9A DFT Frequency 22
4 = 4 pt DFT = 1 [0, 2 ] T = 4 f ' s 8 = 4 = 8 pt DFT /2 f ' s = 2 = 2 [0, ] T = 8 9A DFT Frequency 23
8 = 1 = 8 pt DFT [0, 2 ] 2T = 8 9A DFT Frequency 24
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eferences [1] http://en.wikipedia.org/ [2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003 [3] A graphical interpretation of the DFT and FFT, by Steve Mann