Discrete-Time Fourier Series (DTFS)

Fourier Transforms for Deterministic Processes References

Discrete-time Fourier Series (DTFS)

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 59 Fourier Transforms for Deterministic Processes References Opening remarks

The Fourier series representation for discrete-time signals has some similarities with that of continuous-time signals. Nevertheless, certain di↵erences exist:

I Discrete-time signals are unique over the frequency range f [ 0.5, 0.5) or 2 ]! [ ⇡,⇡) (or any interval of this length). 2 I The period of ?a discrete-time signal is expressed in samples.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 60 Fourier Transforms for Deterministic Processes References Discrete-time signals

I Adiscrete-timesignaloffundamentalperiodN can consist of frequency 1 2 (N 1) components f = , , , besides f =0,theDCcomponent N N ··· N I Therefore, the Fourier series representation of the discrete-time periodic signal contains only N complex exponential basis functions.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 61 Fourier Transforms for Deterministic Processes References Fourier series for d.t. periodic signals

Given a periodic sequence x[k] with period N,theFourierseriesrepresentationforx[k] uses N harmonically related exponential functions

ej2⇡kn/N ,k=0, 1, ,N 1 ··· The Fourier series is expressed as

N 1 j2⇡kn/N x[k]= cne (22) n=0 X

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 62 Fourier Transforms for Deterministic Processes References Fourier coecients and Parseval’s relation

The Fourier coecients c are given by: { n}

N 1 1 j2⇡kn/N c = x[k]e (23) n N Xk=0

Parseval’s result for discrete-time signals provides the decomposition of power in the frequency domain, N 1 N 1 1 P = x[k] 2 = c 2 (24) xx N | | | n| n=0 Xk=0 X

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 63 Fourier Transforms for Deterministic Processes References Power (Line) Spectrum

Thus, we have the line spectrum in frequency domain, as in the continuous-time case.

P [n] P (f )= c 2,n=0, 1, ,N 1 (25) xx , xx n | n| ···

2 th I The term c denotes therefore the power associated with the n frequency | n| component

I The di↵erence between the results in the c.t. and d.t. case is only in the restriction on the number of basis functions in the expansion.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 64 Fourier Transforms for Deterministic Processes References Remarks

I The Fourier coecients c enjoy the conjugate symmetry property { n}

? cn = cN n 1 n =0,N/2 (assuming N is even) (26) 6

I The Fourier coecients c are periodic with the same period as x[k] { n} I The power spectrum of a discrete-time periodic signal is also, therefore, periodic,

Pxx[N + n]=Pxx[n] (27)

I The range 0 n N 1 corresponds to the fundamental frequency range n   0 f = 1 1  n N  N

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 65 Fourier Transforms for Deterministic Processes References Example: Periodic pulse The discrete-time Fourier representation of a periodic signal x[k]= 1, 1, 0, 0 with { } period N =4is given by,

3 1 j2⇡kn/4 1 j2⇡n/4 c = x[k]e = (1 + e ) n =0, 1, 2, 3 n 4 4 Xk=0 This gives the coecients

1 1 1 c = ; c = (1 j); c =0; c = (1 + j) 0 2 1 4 2 3 4

? Observe that c1 = c3.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 66 Fourier Transforms for Deterministic Processes References Power spectrum and auto-covariance function

The power spectrum of a discrete-time periodic signal and its auto-covariance function form a Fourier pair.

N 1 1 j2⇡ln/N P [n]= [l]e (28a) xx N xx l=0 N 1X j2⇡ln/N xx[l]= Pxx[n]e (28b) Xl=0

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 67 Fourier Transforms for Deterministic Processes References Discrete-time Fourier Series

Variant Synthesis / analysis Parseval’s relation (power decomposition) and signal requirements

N 1 N 1 N 1 1 Discrete- x[k]= c ej2⇡kn/N P = x[k] 2 = c 2 n xx N | | | n| Time n=0 k=0 n=0 XN 1 X X Fourier 1 j2⇡kn/N cn , x[k]e x[k] is periodic with fundamental period N Series N Xk=0

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 68 Fourier Transforms for Deterministic Processes References

Discrete-time Fourier Transform (DTFT)

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 69 Fourier Transforms for Deterministic Processes References Opening remarks

I The discrete-time aperiodic signal is treated in the same way as the continuous-time case, i.e., as an extension of the DTFS to the case of periodic signal as N . !1 I Consequently, the frequency axis is a continuum.

I The synthesis equation is now an integral, but still restricted to f [ 1/2, 1/2) or 2 ! [ ⇡,⇡). 2

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 70 Fourier Transforms for Deterministic Processes References Discrete-time Fourier transform (DTFT)

The synthesis and analysis equations are given by:

1/2 1 ⇡ x[k]= X(f)ej2⇡fk df = X(!)ej!k d! (Synthesis) (29) 1/2 2⇡ ⇡ Z Z 1 j2⇡fk X(f)= x[k]e (DTFT) (30) k= X1

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 71 Fourier Transforms for Deterministic Processes References DTFT

Remarks

I The DTFT is unique only in the interval [0, 1) cycles/ sample or [0, 2⇡) rad/sample.

I The DTFT is periodic, i.e., X(f +1)=X(f) or X(! +2⇡)=X(!) (Sampling in time introduces periodicity in frequency) k I Further, the DTFT is also the z-transform of x[k], X(z)= k1= x[k]z , 1 evaluated on the unit circle z = ej! P

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 72 Fourier Transforms for Deterministic Processes References Existence conditions

I The signal should be absolutely convergent, i.e., it should have a ﬁnite 1-norm

1 x[k] < (31) | | 1 k= X1

I Aweakerrequirementisthatthesignalshouldhaveaﬁnite2-norm,inwhichcase the signal is guaranteed to only converge in a sum-squared error sense.

I Essentially signals that exist forever in time, e.g., step, ramp and exponentially growing signals, do not have a Fourier transform.

I On the other hand, all ﬁnite-length, bounded-amplitude signals always have a Fourier transform.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 73 Fourier Transforms for Deterministic Processes References Energy conservation

Energy is preserved under this transformation once again due to Parseval’s relation:

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 74 Fourier Transforms for Deterministic Processes References Energy spectral density

Consequently, the quantity

X(!) 2 S (f)= X(f) 2; S (!)=| | (33) xx | | xx 2⇡ qualiﬁes to be a density function, speciﬁcally as the energy spectral density of x[k].

Given that X(f) is periodic (for real-valued signals), the spectral density of a discrete-time (real-valued) signal is also periodic with the same period.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 75 Fourier Transforms for Deterministic Processes References Example: Discrete-time impulse

The Fourier transform of a discrete-time impulse x[k]=[n] (Kronecker delta) is

1 j2⇡fk X(f)= [n] = [k]e =1 f (34) F{ } 8 k= X1 giving rise to a uniform energy spectral density

S (f)= X(f) 2 =1 f (35) xx | | 8

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 76 Fourier Transforms for Deterministic Processes References Example: Discrete-time impulse

1 2

0.9 1.8

0.8 1.6

0.7 1.4

0.6 1.2

0.5 1

Amplitude 0.4 0.8

0.3 0.6 Energy spectral density 0.2 0.4

0.1 0.2

0 0 −10 −5 0 5 10 −0.4 −0.2 0 0.2 0.4 0.6 Time Frequency (cycles/sample)

(g) Finite-duration pulse (h) Energy spectral density

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 77 Fourier Transforms for Deterministic Processes References Example: Discrete-time ﬁnite-duration pulse Compute the Fourier transform and the energy density spectrum of a ﬁnite-duration rectangular pulse A, 0 k L 1 x[k]=   ( 0 otherwise

Solution: The DTFT of the given signal is

L 1 1 j2⇡fL j2⇡fk j2⇡fk 1 e X(f)= x[k]e = Ae = A 1 e j2⇡f k= k=0 X1 X 1 cos(2⇡fL) S (f)=A2 xx 1 cos 2⇡f

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 78 Fourier Transforms for Deterministic Processes References Example: Discrete-time impulse contd.

100 1 90 0.9 80 0.8 70 0.7 60 0.6

0.5 50

Amplitude 0.4 40

0.3 30 Energy spectral density

0.2 20

0.1 10

0 0 −10 −5 0 5 10 15 20 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Time Frequency (cycles/sample)

(i) Finite-duration pulse (j) Energy spectral density Finite-length pulse and its energy spectral density for A =1,L=10.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 79 Fourier Transforms for Deterministic Processes References Energy spectral density and auto-covariance function

The energy spectral density of a discrete-time aperiodic signal and its auto-covariance function form a Fourier pair.

1 j2⇡lf Sxx(f)= xx[l]e (36a) l= X1 1/2 j2⇡fl xx[l]= Sxx(f)e df (36b) 1/2 Z

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 80 Fourier Transforms for Deterministic Processes References Cross-energy spectral density

In multivariable signal analysis, it is useful to deﬁne a quantity known as cross-energy spectral density,

? Sx2x1 (f)=X2(f)X1 (f) (37)

The cross-spectral density measures the linear relationship between two signals in the frequency domain,whereastheauto-energyspectral density measures linear dependencies within the observations of a signal.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 81 Fourier Transforms for Deterministic Processes References Cross energy spectral density . . . contd.

When x2[k] and x1[k] are the output and input of a linear time-invariant system respectively, i.e.,

n= 1 1 x [k]=G(q )x [k]= g[n]x [k n]=g [k] ?x [k] (38) 2 1 1 1 1 n= X1 two important results emerge

j2⇡f j2⇡f 2 S (f)=G (e )S (f); S (f)= G (e ) S (f) (39) x2x1 1 x1x1 x2x2 | 1 | x1x1

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 82 Fourier Transforms for Deterministic Processes References Discrete-time Fourier Transform

Variant Synthesis / analysis Parseval’s relation (energy decomposition) and signal requirements 1/2 1 1/2 Discrete- x[k]= X(f)ej2⇡fk df E = x[k] 2 = X(f) 2 df xx | | | | Time Z 1/2 k= Z 1/2 X1 Fourier 1 j2⇡fk 1 X(f) x[k]e x[k] is aperiodic; x[k] < or , | | 1 Transform k= k= X1 X1 1 x[k] 2 < (ﬁnite energy, weaker | | 1 k= requirement)X1

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 83 Fourier Transforms for Deterministic Processes References Summary

It is useful to summarize our observations on the spectral characteristics of di↵erent classes of signals.

i. Continuous-time signals have aperiodic spectra ii. Discrete-time signals have periodic spectra iii. Periodic signals have discrete (line) power spectra iv. Aperiodic (ﬁnite energy) signals have continuous energy spectra

Continuous spectra are qualiﬁed by a spectral density function.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 84 Fourier Transforms for Deterministic Processes References Spectral Distribution Function

In all cases, one can deﬁne an energy / power spectral distribution function, (f).

For periodic signals, we have step-like power spectral distribution function, For aperiodic signals, we have a smooth energy spectral distribution function, where one could write the spectral density as,

f Sxx(f)=d(f)/df or xx(f)= Sxx(f) df (40) 1/2 Z .

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 85 Fourier Transforms for Deterministic Processes References

Properties of DTFT

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 86 Fourier Transforms for Deterministic Processes References Linearity property

1. Linearity:

If x [k] F X (!) and x [k] F X (!) then 1 ! 1 2 ! 2

a x [k]+a x [k] F a X (f)+a X (f) 1 1 2 2 ! 1 1 2 2 The Fourier transform of a sum of discrete-time (aperiodic) signals is the respective sum of transforms.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 87 Fourier Transforms for Deterministic Processes References Shift property 2. Time shifting:

If x1[k] F X1(!) then ! j2⇡fD x [k D] F e X (f) 1 ! 1

I Time-shifts result in frequency-domain modulations.

I Note that the energy spectrum of the shifted signal remains unchanged while the phase spectrum shifts by !k at each frequency. Dual: AshiftinfrequencyX(f f ) corresponds to modulation in time, 0 ej2⇡f0kx[k].

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 88 Fourier Transforms for Deterministic Processes References Time reversal

3. Time reversal:

If x[k] F X(!),thenx[ k] F X( f)=X?(f) ! ! If a signal is folded in time, then its power spectrum remains unchanged; however, the phase spectrum undergoes a sign reversal. Dual: The dual is contained in the statement above.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 89 Fourier Transforms for Deterministic Processes References Scaling property

4. Scaling:

If x[k] F X(!) (or x(t) F X(F )), !k ! t then x F X(sf) (or x F X(sF )) s ! s !  ✓ ◆

1 If X(F ) has a center frequency F ,thenscalingthesignalx(t) by a factor c s F results in shifting the center frequency (of the scaled signal) to c s Note: For real-valued functions, it is more appropriate to refer to X(F ) , | |

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 90 Fourier Transforms for Deterministic Processes References Example: Scaling a Morlet wave

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 91 Fourier Transforms for Deterministic Processes References Convolution 5. Convolution Theorem: Convolution in time-domain transforms into a product in the frequency domain. Theorem

If x [k] F X (!) and x [k] F X (!) and 1 ! 1 2 ! 2 1 x[k]=(x ?x )[k]= x [n]x [k n] 1 2 1 2 n= X1 then X(f) x[k] = X (f)X (f) , F{ } 1 2 This is a highly useful result in the analysis of signals and LTI systems or linear ﬁlters.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 92 Fourier Transforms for Deterministic Processes References Product

6. Dual of convolution: Multiplication in time corresponds to convolution in frequency domain.

1/2 x[k]=x1[k]x2[k] F X1()X2(f ) d ! 1/2 Z

I This result is useful in studying Fourier transform of windowed or ﬁnite-length signals such as STFT and discrete Fourier transform (DFT).

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 93 Fourier Transforms for Deterministic Processes References Correlation theorem

7. Correlation Theorem (Wiener-Khinchin theorem for deterministic signals)

Deterministic Fourier Theorem Energy Signal Transform x[k]Magnitude-squared DFT F X(f)

The Fourier transform of the cross-covariance function ] k ) f ( [

x 2 X

XN (fn) ) f

[l] is the cross-energy spectral density ] | | x1x2 ( k [ X x N lim

1 F j2⇡fl xx[l] Wiener-Khinchin Sxx(f) x1x2 [l] = x1x2 [l]e = Sx1x2 (f)=2⇡Sx1x2 (!) Theorem Spectral ACVF F{ } Density l= X1 I This result provides alternative way of computing spectral densities (esp. useful for random signals)

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 94 Fourier Transforms for Deterministic Processes References

Discrete Fourier Transform (DFT) and Periodogram

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 95 Fourier Transforms for Deterministic Processes References Opening remarks

I Signals encountered in reality are not necessarily periodic.

I Computation of DTFT, i.e., the Fourier transform of discrete-time aperiodic signals, presents two diculties in practice: 1. Only ﬁnite-length N measurements are available. 2. DTFT can only be computed at a discrete set of frequencies.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 96 Fourier Transforms for Deterministic Processes References Computing the DTFT: Practical issues

I Can we compute the ﬁnite-length DTFT, i.e., restrict the summation to the extent observed?

I Or do we artiﬁcially extend the signal outside the observed interval? Either way what are the consequences?

I Some form of discretization of the frequency axis, i.e., sampling in frequency is therefore inevitable.

When the DTFT is restricted to the duration of observation and evaluated on a frequency grid, we have the Discrete Fourier Transform (DFT)

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 97 Fourier Transforms for Deterministic Processes References Sampled finite-length DTFT: DFT DFT The discrete Fourier transform of a finite length sequence x[k],k=0, 1, ,N 1 is ··· defined as:

N 1 j2⇡fnk X(fn)= x[k]e , (41) Xk=0 The transform derives its name from the fact that it is now discrete in both time and frequency.

Q: What should be the grid spacing (sampling interval) in frequency?

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 98 Fourier Transforms for Deterministic Processes References Main result

For signal x[k] of length Nl,itsDTFTX(f) is perfectly recoverable from its sampled

version X(fn) if and only if the frequency axis is sampled uniformly at Nl points in [ 1/2, 1/2),i.e.,i↵ 1 2⇡ f = or ! = (42) 4 Nl 4 Nl

See Proakis and Manolakis, (2005) for a proof.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 99 Fourier Transforms for Deterministic Processes References N-point DFT

The resulting DFT is known as the N-point DFT with N = Nl.Theassociatedanalysis and synthesis equations are given by

N 1 j 2⇡ nk X[n] X(f )= x[k]e N n =0, 1, ,N 1 (43a) , n ··· k=0 XN 1 1 j 2⇡ kn x[k]= X[n]e N k =0, 1, ,N 1 (43b) N ··· n=0 X

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 100 Fourier Transforms for Deterministic Processes References Unitary DFT

It is also a common practice to use a factor 1/pN on both (43a) and (43b) to achieve symmetry of expressions.

N 1 1 j2⇡fnk n X[n]= x[k]e f = ,n=0, 1, ,N 1 (44a) p n N ··· N k=0 NX1 1 x[k]= X[n]ej2⇡fnk k =0, 1, ,N 1 (44b) p ··· N n=0 X

The resulting transforms are known as unitary transforms since they are norm-preserving, i.e., x[k] 2 = X[n] 2. || ||2 || ||2

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 101 Fourier Transforms for Deterministic Processes References Reconstructing X(f) from X[n] The reconstruction of X(f) from its N-point DFT is facilitated by the following expression (Proakis and Manolakis, 2005):

N 1 2⇡n 2⇡n X(f)= X P 2⇡f N N (45) N N l n=0 X ✓ ◆ ✓ ◆ sin(⇡fN) j⇡f(N 1) where P (f)= e N sin(⇡f)

I Equation (45) has very close similarities to that for a continuous-time signal x(t) from its samples x[k] (Proakis and Manolakis, 2005). Further, the condition N N is similar to the requirement for avoiding aliasing. I l

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 102 Fourier Transforms for Deterministic Processes References Consequences of sampling the frequency axis When the DTFT is evaluated at N equidistant points in [ ⇡,⇡],oneobtains

2⇡ 1 j2⇡nk/N X n = x[k]e n =0, 1, ,N 1 N ··· ✓ ◆ k= X1 lN+N 1 1 j2⇡nk/N = x[k]e l= k=lN X1 X N 1 1 j2⇡nk/N = x[k lN]e (46) k=0 l= X X1

1 Now, deﬁne x [k]= x[k lN], with period N = N. p p l= 1 Arun K. Tangirala (IIT Madras)X Applied Time-Series Analysis 103 Fourier Transforms for Deterministic Processes References Equivalence between DFT and DTFS Then (46) appears structurally very similar to that of the coecients of a DTFS:

N 1 j2⇡nk/N Ncn = xp[k]e (47) Xk=0

The N-point DFT X[n] of a sequence x = x[0],x[1], ,x[N 1] is equivalent to N { ··· } the coecient cn of the DTFS of the periodic extension of xN .Mathematically,

N 1 1 j 2⇡ kn X[n]=Nc ,c= x[k]e N (48) n n N Xk=0

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 104 Fourier Transforms for Deterministic Processes References Putting together ···

An N-point DFT implictly assumes the given ﬁnite-length signal to be periodic with a period equal to N regardless of the nature of the original signal.

k k I The basis blocks are cos(2⇡ N n) and sin(2⇡ N n) characterized by the index n I The quantity n denotes the number of cycles completed by each basis block for the duration of N samples

I DFT inherits all the properties of DTFT with the convolution property replaced by circular convolution.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 105 Fourier Transforms for Deterministic Processes References DFT: Summary Deﬁnition The N-point DFT and IDFT are given by

N 1 N 1 j2⇡kn/N 1 j2⇡kn/N X[n]= x[k]e ; x[k]= X[n]e N n=0 Xk=0 X

j2⇡/N I Introducing WN = e ,theaboverelationshipsarealsosometimeswrittenas

N 1 N 1 kn 1 kn X[n]= x[k]W ; x[k]= X[n]W N N N n=0 Xk=0 X

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 106 Fourier Transforms for Deterministic Processes References Points to remember

I The frequency resolution in DFT is equal to 1/N or 2⇡/N. Increasing the length artiﬁcially by padding with zeros does not provide any new information but can only provide a better “display” of the spectrum

I DFT is calculated assuming that the given signal x[k] is periodic and therefore it is a Fourier series expansion of x[k] in reality!

I In an N-point DFT, only N/2+1frequencies are unique. For example, in a 1024-point DFT, only 513 frequencies are sucient to reconstruct the original signal.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 107 Fourier Transforms for Deterministic Processes References DFT in practice: FFT

I The linear transformation relationships are useful for short calculations.

I In 1960s, Cooley and Tukey developed an ecient algorithm for fast computation of DFT which revolutionized the world of spectral analysis

I This algorithm and its subsequent variations came to be known as the Fast Fourier Transform (FFT), which is available with almost every computational package. 2 I The FFT algorithm reduced the number of operations from N in regular DFT to the order of N log(N)

I FFT algorithms are fast when N is exactly a power of 2

I Modern algorithms are not bounded by this requirement!

R: fft

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 108 Fourier Transforms for Deterministic Processes References Power or energy spectral density?

I Practically we encounter either ﬁnite-energy aperiodic or stochastic (or mixed) signals, which are characterized by energy and power spectral density, respectively.

I However, the practical situation is that we have a ﬁnite-length signal xN = x[0],x[1], ,x[N 1] . { ··· } I Computing the N-point DFT amounts to treating the underlying inﬁnitely long signal x˜[k] as periodic with period N.

Thus, strictly speaking we have neither densities. Instead DFT always implies a power spectrum (line spectrum) regardless of the nature of underlying signal!

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 109 Fourier Transforms for Deterministic Processes References Periodogram: Heuristic power spectral density

The power spectrum Pxx(fn) for the ﬁnite-length signal xN is obtained as

X[n] 2 P (f )= c 2 = | | (49) xx n | n| N 2 A heuristic power spectral density (power per unit cyclic frequency), known as the periodogram,introducedbySchuster,(1897),fortheﬁnite-lengthsequenceisused,

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 110 Fourier Transforms for Deterministic Processes References Routines in R

Task Routine Remark Convolution convolve, Computes product of DFTs followed by in- conv version (conv from the signal package) Compute IR impz Part of the signal package Compute FRF freqz Part of the signal package DFT fft Implements the FFT algorithm Periodogram spec.pgram, Part of the stats and TSA packages, respec- periodogram tively

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 111 Fourier Transforms for Deterministic Processes References Bibliography I

Antoniou, A. (2006). Digital Signal Processing: Signals Systems and Filters. USA: McGraw-Hill. Bloomﬁeld, P. (2000). Fourier Analysis of Time Series: An Introduction. 2nd edition. New York, USA: John Wiley & Sons, Inc. Cohen, L. (1994). Time Frequency Analysis: Theory and Applications. Upper Saddle River, New Jersey, USA: Prentice Hall. Hamilton, J. D. (1994). Time Series Analysis. Princeton, NJ, USA: Princeton University Press. Lighthill, M. (1958). Introduction to Fourier Analysis and Generalized Functions. Cambridge, UK: Cam- bridge University Press. Priestley, M. B. (1981). Spectral Analysis and Time Series. London, UK: Academic Press. Proakis, J. and D. Manolakis (2005). Digital Signal Processing - Principles, Algorithms and Applications. New Jersey, USA: Prentice Hall.

Arun K. Tangirala (IIT Madras) Applied Time-Series Analysis 112 Fourier Transforms for Deterministic Processes References Bibliography II

Schuster, A. (1897). “On lunar and solar periodicities of earthquakes”. In: Proceedings of the Royal Society 61, pp. 455–465. Shumway, R. and D. Sto↵er (2006). Time Series Analysis and its Applications. New York, USA: Springer- Verlag. Smith, S. W. (1997). Scientist and Engineer’s Guide to Digital Signal Processing. San Diego, CA, USA: California Technical publishing. Tangirala, A. K. (2014). Principles of System Identiﬁcation: Theory and Practice. Boca Raton, FL, USA: CRC Press, Taylor & Francis Group.

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