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# Discrete Fourier Transform

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### Discrete Fourier Transform

1. 1. Discrete Fourier Transform (DFT) Presented by: SHAHRYAR ALI
2. 2. Discrete Time FT (DTFT) DTFT defined as: Note: continuous frequency domain! (frequency density function) s si +∞ ly naa S(f) = ∑ s[n] ⋅ e − j 2 π f n n = −∞ Holds for Aperiodic signals is hes 2π nt 1sy s[n] = ⋅ ∫ S(f)e j 2 π f ndf 2π 0
3. 3. Problem With DTFT– Defined for infinite-length sequences.– From numerical computation viewpoint: “It is troublesome as one has to evaluate infinite sums at uncountable infinite frequencies”– To use Matlab, we have to truncate sequences and then evaluate the expression at many finite points.
4. 4. Therefore:• We turn our attention to a numerically computable transform.• It is obtained by sampling the DTFT transform in the frequency domain (or the z-transform on the unit circle).But..• We know that a periodic function can always be represented by: “A linear combination of harmonically related complex exponentials”
5. 5. The Discrete Fourier Series• So we have Discrete Fourier Series representation.• Definition: Periodic sequence. ~ (n) = ~ (n + kN ), ∀n, k x xN: the fundamental period of the sequences
6. 6. Discrete Fourier SeriesAnalysis equation: ~ N −1 ~[n]e − j( 2 π / N)kn X[k ] = ∑x n=0Synthesis equation: ~[n] = 1 N −1 ~ x ∑ X[k ]e j( 2 π / N)kn N k =0
7. 7. • For convenience we sometimes use: − j( 2 π / N ) WN = eSo.. ~ N −1 ~[n]Wkn X[k ] = ∑x N n=0 ~ { X ( K ), k = 0,±1, } called the discrete Fourier series are coefficients. ~[n] = 1 N −1 ~ x ∑ X[k ]WN kn − N k =0
8. 8. Properties of DFS• Linearity ~ [n] ~ x1 ← DFS →   X1 [k ] ~ [n] ~ x 2 ← DFS →   X2 [k ] ~ ~ a~1 [n] + b~2 [n] x x ← DFS → aX1 [k ] + bX2 [k ]  • Shift of a Sequence ~[n] ~ x ← DFS →  X[k ] ~ X[n] ← DFS → N~[ − k ]  x• Duality ~[n] ~ x DFS ←  →  X[k ] ~[n − m] ~ x ←  → e − j2 πkm / NX[k ] DFS  e j2 πnm / N~ x [n] ← → ~[k − m] DFS  X
9. 9. The Fourier Transform of Periodic Signals• Periodic sequences are not absolute or square summable – Hence they don’t have a Fourier Transform.• We can represent them as sums of complex exponentials: DFS.• We can define a periodic signal whose primary shape is that of the finite duration signal .• We then use the DFS on this periodic signal.• So we define a new transform called the Discrete Fourier Transform (DFT), which is the primary Period of the DFS.
10. 10. Discrete Fourier Transform (DFT)
11. 11. Discrete Fourier Transform (DFT)• Discrete Fourier transform (or DFT) takes a finite number of samples of a signal.• It then transforms them into a finite number of frequency samples .• The discrete Fourier transform does not act on signals that exist at all time.• The DFT can be used in practice using a fast Fourier transform (FFT) algorithm.
12. 12. Fourier analysis Input Time Signal Frequency spectrum2.5 21.5 10.5 0 0 1 2 3 4 5 6 7 8 Periodic FS Discrete time, t Continuous (period T)2.5 21.5 Aperiodic FT Continuous 10.5 0 0 2 4 6 8 10 12 time, t 2.5 2 DFS Discrete 1.5 Periodic 1 0.5 0 (period T) 0 1 2 3 4 5 6 7 8 time, tk Discrete DTFT Continuous 2.5 2 Aperiodic 1.5 1 0.5 0 time, tk DFT 0 2 4 6 8 10 12 Discrete
13. 13. Discrete Fourier Transform (DFT)Definition: The Discrete Fourier Transform (DFT) is defined by: Where n = 0, 1, 2, …., N-1 The Inverse Discrete Fourier Transform (IDFT) is defined by: where k = 0, 1, 2, …., N-1. Same form of DFS but for aperiodic signals. Signal treated as periodic for computational purpose only.
14. 14. Sample X at N points O<w<2π x(2) x(1) x(o) w x(N-1)
15. 15. DFT at work• To see how DFT equation actually works in practice, let’s do a simple example - calculate DFT of 4 element sequence, x(n)={1,1,0,0} for k=0 4−1X ( 0 ) = ∑ x ( n ) e− j 2π ×0×n 4 n= 0 = x ( 0 ) e − j 2π ×0×0 4 + x ( 1) e − j 2π ×0×1 4 + x ( 2 ) e− j 2π ×0×2 4 + x ( 3) e − j 2π ×0×3 4 = 1×e− j 2π ×0×0 4 + 1×e− j 2π ×0×1 4 + 0 ×e − j 2π ×0×2 4 + 0 ×e− j 2π ×0×3 4 =2
16. 16. DFT at work for k=1X ( 1) = x ( 0 ) e − j 2π ××0 4 + x ( 1) e − j 2π ×× 4 + x ( 2 ) e − j 2π ××2 4 + x ( 3) e − j 2π ××3 4 1 11 1 1 = 1×e − j 2π ××0 4 + 1 ×e − j 2π ×× 4 + 0 ×e − j 2π ××2 4 + 0 ×e − j 2π ××3 4 1 11 1 1  π   π  = 1 +  cos  ÷− j sin  ÷÷  2  2  = 1− j• Following the same procedure we also get: X ( 2) = 0 X ( 3) = 1 + j• The result: DFT({1,1,0,0})={2,1+j,0,1-j}
17. 17. DFT Properties Time FrequencyLinearity a·s[n] + b·u[n] a·S(k)+b·U(k) 1 N−1Multiplication s[n] ·u[n] ⋅ ∑S(h)U(k - h) N h =0 N− 1Convolution S(k)·U(k) ∑ s[m] ⋅ u[n − m] m= 0Time shifting s[n - m] 2π k ⋅m −j e T ⋅ S(k)Frequency shifting S(k - h) 2π h t +j e T ⋅ s[n]
18. 18. s[n] S(f) (a) (b) 0 T/2 T 2T f s”[n] IDFT DFT(c) (d) (e) (f) cK (a) Aperiodic discrete signal. (b) DTFT transform magnitude. (c) Periodic version of (a). (d) DFS coefficients = samples of (b). (e) Inverse DFT estimates a single period of s[n] (f) DFT estimates a single period of (d).
19. 19. Why DFT is important? To find the frequency content of a signal. • To design an audio format (e.g., CD audio). • To design a communications system (what bandwidth is required?). To determine the frequency response of a structure. • A musical instrument.
20. 20. The Fast Fourier Transform• The fast Fourier transform (FFT) is simply a class of special algorithms which implement the discrete Fourier transform .• It calculates with considerable savings in computational time.• Maximum efficiency of computation is obtained by constraining the points to be an integer power of two, e.g. 1024 or 2048.
21. 21. QUESTIONS???