The Discrete Fourier Transform (DFT) Properties of the DFT DFT-Specic Properties Power spectrum estimate. Alex Sheremet.

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1 4. April 2, 27 -order sequences Measurements produce sequences of numbers Measurement purpose: characterize a stochastic process. Example: Process: water surface elevation as a function of time Parameters: mean water level, wave variance (RMS height), power spectrum etc... Commonly obtainable data are: not one realization; not a nite segment of one realization; but a discrete nite segment of one realization. Measurements = nite sequence of numbers. Goal : Apply the theory (e.g. spectral analysis) to sequences.

2 Denition & properties -order sequences -order sequence is: a set of numbers: {g k }, g k C, k =,,...,. 2 an -dimensional vector: g = (g, g 2,..., g ) C 3 a complex function g of an integer argument g(k). Term-by-term operations op set function vector + {a k } + {b k } = {a k + b k } a + b a + b {a k } {b k } = {a k b k } ab? α α {a k } = {αa k } αa αa / {a k } / {b k } = {a k /b k } a/b?? means no commonly used corresponding op. α is a complex constant Periodicity and symmetry -order sequences As a function, g is dened for k =,,...,. Extend k by gluing together copies of g. Then Extended g is -periodic g(k + ) = g(k). 2 Can dene symmetries for g continuous g( t) = ±g(t) discrete g( k) = g( k) = ±g(k) Using sequence notation g k = g k = ±g k

3 Example: Periodicity -order sequences k = 3; = ; k + = 3; k = 7. Example: Even/Odd -order sequences k = 3; = ; k + = 3; k = 7; k = 7

4 -order sequences Problem: Fourier representation for sequences Measured data is discrete: function g of integer argument, g(k), k =,,...,. Is it possible to build a Fourier representation for g? (DFT = Discrete Fourier Transform) What would be it's properties? What properties of the continuous version transfer to DFT? What new properties would DFT have? Procedure for building DFT -order sequences eeded Continuous Discrete time/freq. domain Fourier pair axis g(t) G(f ) = ĝ(f ) grids g n = g(n) G k = G(k) = ĝ(k) g ĝ Remarks: G(f ) = g(t)e 2πift dt g(ft) = G(t)e2πift df G = D [g] g = D [G] The goal is to dene and study DFT operator D; 2 Grids do not enter DFT process: Djust maps sequence sequence; 3 Grids important for physical intrerpretation; 4 Will prefer G(k) instead of ĝ(k);

5 Step : Discretize FT -order sequences Discretize Fourier Integrals: Direct: Time grid:, t n = n t, n =,,..., t ; G(f ) = g(t) e 2πift dt t n= Time grid dened. Frequency grid OT dened. g(t n )e 2πift n t 2 Inverse: Freq. grid: f k = k f, k =,,..., f ; g(t) = G(f ) e 2πift df f k= Frequency grid dened. Time grid OT dened. o visible relation between time-freq. grids. Grids important for physical intrerpretation. G(f k )e 2πif k t f Step 2: Dene grids -order sequences Time/freq grids have to satisfy Invertibility: t = f =. DFT = linear transformation of vectors. Physical constraints: Time grid: dened by measurement. points t n = n t, n =,,..., ; T = ( ) t; 2 Frequency grid dened by physics: largest period = T f = T ; highest frequency on grid: f max = ( ) f = T ; points; f k = k f, k =,,..., ; f = /T ;

6 Step 3: Put it all together -order sequences Time grid: t n = n t, n =,,..., ; T = ( ) t; Frequency grid: f k = k f, k =,,..., f ; f = /T ; f k t n = kn f t = kn ote: G(f k ) g(t n ) T T k= n= g(t n )e 2πif k t n = T G(f k )e 2πi kn n= g n e 2πi in G(f k ), rst & last terms: g & g (no exponential); If g = g (periodic sequence), rename =. kn Recount for a periodic sequence -order sequences If the sequence is periodic, we only need terms. Renaming : t = T t = T 2 n= n= 3 {g n } has terms 4 but g = g (periodic) Renaming

7 Denition and grids for DFT -order sequences The DFT form below is from J Weaver (Mathematica). Denition Grids: t n = n t, t = T /; f k = k f ; f = /T ; 2 Grid reciprocity: t f = /, and Tf max =. 3 DFT and IDFT are dened as: G k = n= g n e 2πif k t n = n= g n e 2πi kn = n= g n ( Ω kn ), g n = j= G k e 2πif k t n = k= G k e 2πi kn = k= G k Ω kn. 4 Ω = e 2πi is called DFT kernel. Other DFT forms -order sequences See for example Briggs & Henson The DFT, an owner's manual Mathematica Matlab IMSL G k = P n= g kn 2πi P ne g n = k= G kn 2πi ke G k = P n= g (k )(n ) 2πi ne g n = P k= G (k )(n ) 2πi ke G k+ = P n= g kn 2πi P n+e g n+ = k= G kn 2πi k+e etc...

8 Matricial form of DFT -order sequences Use the vector notation g = {g n } and G = {G k }; 2 introduce the matrix Ω = ( ) Ω kn, k, n =,... ; Sequence Matrix G k = g k = P n G k P n g `Ωkn n `Ωkn G = Ω g g = Ω G where Ω = exp(2πi/). For example: G G G 2... G C A = Ω Ω 2... Ω Ω 2 Ω 4... Ω 2( ) Ω Ω 2( )... Ω ( )2 C A g g g 2... g C A -order sequences DFT Kernel matrix is orthogonal (unitary) Fourier basis functions are orthonormal: e2πift dt = δ(f ). What about the DFT? Rows u k = {( ) } { } u k = Ω kn of kernel matrix Ω : Compare with Theorem u k = ( u k = ( n e 2πik e 2πik 2... e 2πik e 2πif k t e 2πif k t 2... e 2πif k t U(t n ) = e 2πift n. For any integer k, with n and n, n= ( ) u k = n n= Ω kn = 2πi kn e =. n= ) )

9 DFT Kernel orthonormal -order sequences Proof. Form the sequence V n = V n+ V n = Ωk(n+) Ω k Ωkn Ω k, Ωkn Ω k = Ωkn ( Ω k ) Ω k V n primitive of Ω kn ; V n is periodic, V n = V n+. = Ω kn. n= Ω kn = n= V n = V V =. DFT as operator Convolution, Correlation, Parseval Relation {G k } = D [{g n }] ; {g n } = D [{G k }] D [{g n }]] = n= g n Ω kn. otations f, g, {f j },{g k } are used for sequences the DFT are capitalized F, G A term in the sequence is denoted by the name of the sequence with an index f j, g k {G k } = k= g n (Ω k ) n G = D [g]

10 DFT: The usual properties () Convolution, Correlation, Parseval Relation Linearity: D [af + bg] = af + bg. Shift : FT F[X (t a)] = ˆX (f ) e 2πifa DFT n D [{g n m }] = {G k } Ω kn o Shift 2: F[X (t) e 2πif t ] = ˆX (f f ) D [{g n } {Ω mn }] = {G k m } Derivative : Derivative 2: For example, d ˆX (f ) df» dx (t) F dt h n = 2πiF [tx (t)] {G j } = D {g k } Ω k oi o = 2πif ˆX (f ) D [ {gk }] = {G j } nω j D [{f k } {cos(2πkn/)}] = 2 [{F j+n} + {F j n }] D [{f k } {sin(2πkn/)}] = i 2 [{F j+n} {F j n }] DFT: The usual properties (3) Convolution, Correlation, Parseval Relation Symmetry relations g real, even real, odd imaginary, even imaginary, odd G real, even imaginary, odd imaginary, even real, odd

11 Plots Convolution, Correlation, Parseval Relation.8 (a) (b).8.8 (a) (b) (c) (d).5.5 (c) (d) t f t f Examples Convolution, Correlation, Parseval Relation th-order Delta sequence: with k =,,...,, {δ k } = {,,..., }, Unit sequence, with k =,,...,,{u k } = {,,..., }, Delta and Unit sequences are a DFT pair: {u j} = D [{δ k }]. {δ j } = D [{u k }]. 2 The DFT of the cosine is the sum of 2 symmetric deltas: [{ ( )}] 2πkn D cos = 2 [{δ j+n} + {δ j n }].

12 Convolution Convolution, Correlation, Parseval Relation Let f and g be two -order sequences, Denition Theorem {(f g) n } = m= {f m } {g n m }. (i) D [fg] = D [f ] D [g], (ii) D [f g] = D [f ] D [g], Correlation Convolution, Correlation, Parseval Relation Denition Theorem {(f g) n } = m= {f m } { g n+m}. (i) D [fg ] = F G, (ii) D [f g] = F G,

13 Parseval Relation Convolution, Correlation, Parseval Relation Theorem If Let f and F form a DFT pair of th-order sequences, k= f n 2 = k= F k 2 DFT-specic properties Zero padding Aliasing What happens when a sequence is padded with zeros? Use symmetric DFT {G j } = D [{g k }] ; D[ ] = + /2 k= /2 [ ] Ω jk. g old -order sequence; g new M-order sequence zero-padded g new k = M/2 k /2 g old /2 k /2 k /2 k M/2

14 Zero padding Zero padding Aliasing If say M + = p( + ), DFT of g new : G old j = pg new pj, j = /2,..., /2. Reciprocity relations : f old = p f new. Zero-padding is a technique which renes the frequency grid. Zero padding Zero padding Aliasing (a) (a) k k (b) (b) j j a) sequence g; b) its Fourier pair G. Blue line: true shape of g and G. Red circles: zero-padded. Blue stars: no zero-padding.

15 Aliasing Zero padding Aliasing Aliasing: misrepresenting an oscillation which cannot be resolved as a dierent (lower) frequency k Denition The frequency f y = f max = is called the yquist frequency. 2 2 t 2 f y is the highest frequency that can be resolved un-aliased by spectral analysis from a sequence samplet at t. Shannon Sampling Theorem Zero padding Aliasing Theorem (Shannon sampling theorem) ĝ(f ) = for f > f y. If t is chosen so that t fmax, then g(t) may be reconstructed exactly from its samples g n = g(n t) by g(t) = n= g n sinc [ π (t tn ) t ].

16 Continuous Discrete X (t) = one realization; 2 DFT pair ˆX = F [X ] ; 3 Sample spectrum T ˆX T (f ) 2 ; { 4 h(f ) = E lim T ˆX T (f ) T 2 } {g k } = one realization; 2 DFT pair {G n } = D [{g k }] ; { 3 Periodogram G n 2} ; 4 Average, let. Step 3: Periodogram Parseval, continuous: Parseval, discrete: X (t) 2 dt = ˆX (f ) 2 df k= g n 2 = k= G k 2 Denition Sample spectrum (periodogram) of sequence{g n } is the sequence{p k } : P k = G x 2 Periodogram = natural estimate of power spectrum: {P k } = DFT of autocorrelation {c n } = n= {g m} { } gn+m 2 Direct discretization of continuous version of denition.

17 Step 4: Let... {P k } is a consistent estimate of the power spectrum if a) lim E [P k ] = h(f k ); and b) lim σp 2 = Example: White noise process. Left: {P k } for =2, 5,, 5. Right: variance of P k versus (a) (b) I k I k (c) (d) 4.5 I k I k 2 variance k k It does not work! The raw periodogram is extremely poor as estimate h(f k ): P k is not a consistent estimate: σp 2 2 P k = P(f k ) is wildly uctuating. as. ote: we did not do any average yet!

18 Periodogram statistics Theorem If g is a discrete 2-order stationary process, g = (, σg), 2 then G = D [g] = (, σ2 g) (µ G = ; σ 2 = G σ2 g ; 2 P k and P k+m are independent for m, and σ 2 = P 4σ4 g, for n. Increasing ncreases frequency resolution but does not reduce oscillations: P k are independent does not reduce errors: σp 2 independent of. Heuristical explanation Problem: Error (variance) of auto-correlation{c n } decays like / as. 2 {P k } = D [{c n }] = linear combination of auto-correlations 3 cumulative eects of /-like terms is order. Fix: Since the c n as n, end points contribute only error! Truncate the auto-correlation (maximum lag) Window the time series! Smooth the periodogram!

19 Discarding large lags Windowing Typical autocorrelation (blue line) curve and associated variance (read: error bars and light blue band). Excluding large lags windowing the time series by a window of T length. Bartlett procedure: average over time blocks Bartlett: goal is to reduce variance Split {g n }, n =,,..., into M sub-blocks of{g n } m of /M points. 2 Calculate autocovariance {c n } m for each sub-block m =,... M. 3 Calculate periodogram {P k } m = D [{c n } m ] for each sub-block 4 Average periodogram over sub-blocks: {H k } = M m {P k} m. The procedure is equivalent to windowing by triangular (Bartlett) window.

20 Welch's direct method Compute DFT of the time series rather than DFT of the autocovariance: Divide{g n } into M sub-blocks of{g n } m of K = /M points; sub-blocks allowed to overlap (typically by 5 %). 2 Window each sub-block: g n W = {gn } {w n }, {w n }= window. Redundancy due to overlapping is reduced by window tapering. 3 Tapering reduces the variance (see next slide). Adjust for variance change due to windowing: {g w n } m {g w n } m q P ˆ{gn } m 2/q P ˆ{g w n } m 2. 4 DFT each tapered sub-block, calculate the periodogram : {P k } m = K G w k 2. 5 Average over sub-blocks: P {H k } = M m {P k} m. Tapering reduces variance Windowed signal (red) has smaler variance than the original signal (blue).

21 Spectrum 6 power density (a) 25 2 (b) power density frequency (Hz) a) {P k } = G k 2, calculated using the entire time series; b) Spectrum estimate by Welch's direct method.

Fourier Analysis Linear transformations and lters. 3. Fourier Analysis. Alex Sheremet. April 11, 2007

Fourier Analysis Linear transformations and lters. 3. Fourier Analysis. Alex Sheremet. April 11, 2007 Stochastic processes review 3. Data Analysis Techniques in Oceanography OCP668 April, 27 Stochastic processes review Denition Fixed ζ = ζ : Function X (t) = X (t, ζ). Fixed t = t: Random Variable X (ζ)