Computation of Phasor from Discrete Fourier Transform

Home , Phasor

S. A. Soman

Department of Electrical Engineering, IIT Bombay Computation of Phasor from Discrete Fourier Transform

Outline

1 DFT of a Sinusoid

2 Phase Computation using DFT

3 Full Cycle Recursive DFT

4 Half Cycle Recursive Form for Phase Estimation

5 Half Cycle Recursive DFT Computation of Phasor from Discrete Fourier Transform DFT of a Sinusoid

DFT of a Sinusoid

Consider a sinusoidal input signal of frequency ω0, given by √ x(t) = 2X cos (ω0t + φ) (1)

This signal is conveniently represented by a phasor X¯

X¯ = Xejφ = X(cos φ + j sin φ) (2)

Assume that x(t) is sampled N times per cycle such that T0 = Nts. √ 2π x = 2X cos k + φ (3) k N Computation of Phasor from Discrete Fourier Transform DFT of a Sinusoid

The Discrete Fourier Transform of xk contains the fundamental frequency component given by

N−1 2 X −j 2π k X = x e N = X − jX (4) 1 N k c s k=0 Where, N−1 2 X 2π X = x cos k (5) c N k N k=0 N−1 2 X 2π X = x sin k (6) s N k N k=0 Computation of Phasor from Discrete Fourier Transform DFT of a Sinusoid

Substituting xk from (3) in (5) and (6) it can be shown that for a sinusoidal input signal given by (1) √ √ Xc = 2X cos φ Xs = − 2X sin φ (7)

From equation (2), (4) and (7) it follows that

¯ 1 1 X = √ X1 = √ (Xc + jXs) (8) 2 2 When the input signal contains other frequency components as well, the phasor calculated by equation (8) is a ﬁltered fundamental frequency phasor. Computation of Phasor from Discrete Fourier Transform Phase Computation using DFT

Phase Computation using DFT

w w Let Xc and Xs indicate Xc and Xs component of DFT for w th window. Window number is the sample number of the ﬁrst sample in the active window.

N−1 2 X 2π √ 2π X w = x cos k = 2X cos w + φ c N k+w N N k=0

Similarly,

N−1 2 X 2π √ 2π X w = x sin k = − 2X sin w + φ s N k+w N N k=0 Computation of Phasor from Discrete Fourier Transform Phase Computation using DFT

DFT estimated at w th window for (m = 1) is given by

√ 2π 2π X w = X w −jX w = 2X cos w + φ + j sin w + φ 1 c s N N

Phasor estimate for w th window is given by

¯ w 1 w w jφ j 2π w X = √ (Xc + jXs ) = Xe e N 2

j 2π The phasor estimate rotates at a speed of e N per window which can be directly derived from DFT phase shifting property. Computation of Phasor from Discrete Fourier Transform Phase Computation using DFT

In the computation of phasor, we would not like this phasor rotation to occur due to DFT phase shifting. This can be avoided by

w jφ j 2π w X¯ = Xe e N

w −j 2π w jφ X¯ e N = Xe

N−1 ! 2 X −j 2π k −j 2π w jφ x e N e N = Xe N k+w k=0

w+N−1 2 X −j 2π k jφ x e N = Xe N k k=w Computation of Phasor from Discrete Fourier Transform Full Cycle Recursive DFT

Full Cycle Recursive DFT

From equation (5) and (6),with elimination of phasor rotation we have

N+w−1 2 X 2πk X w = x cos c N k N k=w

N+w−1 2 X 2πk X w = x sin s N k N k=w w The computation for Xc can be visualized from the following ﬁgure. Computation of Phasor from Discrete Fourier Transform Full Cycle Recursive DFT

Figure: Mechanism of Recursive DFT Computation of Phasor from Discrete Fourier Transform Full Cycle Recursive DFT

It is now obvious, that

2π X w+1 = X w + (x − x ) cos w (9) c c w+N w N

Similarly

2π X w+1 = X w + (x − x ) sin w (10) s s w+N w N

It reduces computation from 2N multiply add operation in normal DFT to 4 additions and 2 multiplications per update. Computation of Phasor from Discrete Fourier Transform Full Cycle Recursive DFT

Question

If we follow the convention that latest sample number corresponds to window number then show that in recursive full cycle fourier algorithm update equations are given by

2π X w+1 = X w + (x − x ) cos w c c w w−N N 2π X w+1 = X w + (x − x ) sin w s s w w−N N Computation of Phasor from Discrete Fourier Transform Half Cycle Recursive Form for Phase Estimation

Half Cycle Recursive Form for Phase Estimation

Half cycle form of DFT phasor estimation is given by

N+w−1 2 X π X w = x cos k (11) c N k N k=w N+w−1 2 X π X w = x sin k (12) s N k N k=w Question √ π By substituting x = 2X cos k + φ validate eq (11) and k N (12). Show that √ √ w w Xc = 2X cos φ Xs = − 2X sin φ (13) Computation of Phasor from Discrete Fourier Transform Half Cycle Recursive DFT

Half Cycle Recursive DFT

Recursive form of half cycle DFT can be derived in an analogous manner to full cycle DFT. Recursive update forms for fundamental phasor computation π π X w+1 = X w + x cos (N + w) − x cos w c c w+N N w N π = X w − (x + x ) cos w c w+N w N π π X w+1 = X w + x sin (N + w) − x sin w s c w+N N w N π = X w − (x + x ) sin w s w+N w N Computation of Phasor from Discrete Fourier Transform Half Cycle Recursive DFT

Question

If we follow the convention that latest sample number corresponds to window number then show that in recursive half cycle fourier algorithm update equations are given by π X w+1 = X w + (x − x ) cos w c c w w−N N π X w+1 = X w + (x + x ) sin w s s w w−N N Computation of Phasor from Discrete Fourier Transform Half Cycle Recursive DFT

Discussion

Can we implement Recursive form on a DSP or a microprocessor? Computation of Phasor from Discrete Fourier Transform Half Cycle Recursive DFT

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