Signals and Systems - ETH Z€¦ · Discrete Fourier series representation of a periodic signal...

Signals and Systems

Lecture 5: Discrete Fourier Series

Dr. Guillaume Ducard

Fall 2018

based on materials from: Prof. Dr. Raffaello D’Andrea

Institute for Dynamic Systems and Control

ETH Zurich, Switzerland

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Outline

1 The Discrete Fourier SeriesDiscrete Fourier series representation of a periodic signalProperties of the discrete Fourier seriesDFS coefficients of real signals

2 Response to Complex Exponential SequencesComplex exponential as inputDFS coefficients of inputs and outputs

3 Relation between DFS and the DT Fourier TransformDefinitionExample

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The Discrete Fourier SeriesResponse to Complex Exponential Sequences

Relation between DFS and the DT Fourier Transform

Discrete Fourier series representation of a periodic signalProperties of the discrete Fourier seriesDFS coefficients of real signals

Overview of Frequency Domain Analysis in Lectures 4 - 6

Tools for analysis of signals and systems in frequency domain:

The DT Fourier transform (FT): For general, infinitely long andabsolutely summable signals.⇒ Useful for theory and LTI system analysis.

The discrete Fourier series (DFS): For infinitely long butperiodic signals⇒ basis for the discrete Fourier transform.

The discrete Fourier transform (DFT): For general, finitelength signals.⇒ Used in practice with signals from experiments.

Underlying these three concepts is the decomposition of signals intosums of sinusoids (or complex exponentials).

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Outline

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Recall on periodic signals

A periodic signal displays a pattern that repeats itself, for exampleover time or space.

Recall

A periodic sequence x with period N is such that

x[n+N ] = x[n], ∀n

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Overview of the Discrete Fourier Series

Analysis equation (Discrete Fourier Series)

DFS coefficients are obtained as:

X[k] =

N−1∑

x[n]e−jk 2πN

Synthesis equation (Inverse Discrete Fourier Series)

x[n] =1

N−1∑

X[k]ejk2πN

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Discrete Fourier series representation of a periodic signal

The Discrete Fourier Series (DFS) is an alternativerepresentation of a periodic sequence x with period N .

The periodic sequence x can be represented

as a sum of N complex exponentials with frequencies k 2πN, where

k = 0, 1, . . . , N−1:

x[n] =1

N−1∑

X[k]ejk2πN

for all times n, where X[k] ∈ C is the kth DFS coefficient

corresponding to the complex exponential sequence ejk2πN

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Discrete Fourier series representation of a periodic signal

Remarks:

1 The frequency 2πN

is called the fundamental frequency and isthe lowest frequency component in the signal. (We will latershow that there is no loss of information in thisrepresentation.)

2 We only need N complex exponentials to represent a DTperiodic signal with period N .Indeed, there are only N distinct complex exponentials withfrequencies that are integer multiples of 2π

ej(k+N) 2πN

n = ejk2πN

nejN2πN

n = ejk2πN

nej2πn = ejk2πN

3 graphical representation (shown during class).

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Discrete Fourier series coefficients

The DFS coefficients of Equation (1) are obtained from theperiodic signal x (the role of X and x are permuted with a minus sign in

the exponential) as:

X[k] =N−1∑

x[n]e−jk 2πN

The DFS operator is denoted as Fs , where:

X = Fsx

and x = F−1s X.

The pairs are usually denoted as x[n] ←→ X[k].

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Discrete Fourier series coefficients

Note that, like the underlying sequence x, X is periodic withperiod N :Proof:

X[k +N ] =

N−1∑

x[n]e−j(k+N) 2πN

N−1∑

x[n]e−jk 2πN

n e−j2πn︸︷︷︸

=1∀n

= X[k].

ConclusionsWhen working with the DFS, it is therefore common practice toonly consider one period of the sequence X[k], that is: only theN DFS coefficients as

X[k] for k = 0, 1, . . . , N−1.10 / 27

Proof that the DFS operator in invertible - Part I

The following identity highlights the orthogonality of complex exponentials:

N−1∑

ej(r−k) 2π

1 for r − k = mN, m ∈ Z

0 otherwise.

Proof:Case 1 : r − k = mN . In this case, we have

ejmN 2π

Nn = e

j2πmn = 1 for all m,n

N−1∑

ej(r−k) 2π

N−1∑

NN= 1.

Case 2 : r − k 6= mN . Define l := r − k. We then have

N−1∑

ejl 2π

1− ejl2πN

the above equation is a geometric series and ejl2πN 6= 1 since l 6= mN . For

l 6= mN , we therefore have 1N

1−ej2πl

1−ejl 2π

= 0.11 / 27

Proof that the DFS operator in invertible - Part II

In order to prove that F−1s is the inverse transform of Fs , we need to show:

1 FsF−1s = I

2 and F−1s Fs = I , where I is the identity operator.

We now show that FsF−1s = I :

FsF−1s X[k] =

N−1∑

X[r]ejr2πN

e−jk 2π

N−1∑

ej(r−k) 2π

︸︷︷︸

From above equation, the term in underbrace is equal to 1 for r = k mod N

and 0 otherwise. Thus:

FsF−1s X[k] =

N−1∑

ej(r−k) 2π

= X[k mod N ] = X[k].

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In a similar way, it can also be shown that F−1s Fs = I.

Remark:As both x and X are periodic with period N , we can sum over anyN consecutive values (denoted as 〈N〉):

x[n] =1

k=〈N〉

X[k]ejk2πN

n and X[k] =∑

n=〈N〉

x[n]e−jk 2πN

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Outline

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Properties of the discrete Fourier series

Linearity

a1x1[n]+ a2x2[n] ←→ a1X1[k]+ a2X2[k]

Parseval’s theorem (recall: period is N)

N−1∑

|x[n]|2 =1

N−1∑

|X[k]|2

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Proof of Parseval’s theorem

N−1∑

|X[k]|2 =?

N−1∑

X∗[k]X[k] (where ∗ denotes the complex conjugate)

N−1∑

(N−1∑

x∗[n]ejk

N−1∑

x∗[n]X[k]ejk

=N−1∑

x∗[n]

N−1∑

X[k]ejk2πN

︸︷︷︸

N−1∑

x∗[n]x[n] =

N−1∑

|x[n]|2

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Outline

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DFS coefficients of real signals: x[n] ∈ R for all n1 So far: x[n] ∈ C for all n.

2 Now we consider: x[n] ∈ R for all n; (most often case in practice)

Recall: DFS coefficients of a periodic signal x : X[k] =N−1∑

x[n]e−jk 2πN

Letting k = N − γ, where γ is an integer, leads to

X[N − γ] =N−1∑

x[n]e−j(N−γ) 2πN

N−1∑

x[n] e−j2πn

︸︷︷︸=1∀n

ejγ 2π

N−1∑

x[n]ejγ2πN

N−1∑

x∗[n]ejγ

n = X∗[γ],

We used that for a real signal x, x[n] = x∗[n].

Conclusions:

For a real signal x, we have that X[N − k] = X∗[k].

Take γ = 0 → X[N ] = X∗[0], γ = N → X[0] = X∗[N ].By periodicity X[0] = X[N ]. Thus X[0] = X∗[0].

For a real signal, X[0] is therefore always real.18 / 27

DFS coefficients of real signals

Example

Consider the periodic signal :

x = . . . ,1

2, 1, . . . ,

Questions:

1 What is the period of such signal ?

2 Compute the Discrete Fourier Series coefficients.

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Complex exponential as inputDFS coefficients of inputs and outputs

Outline

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Consider a periodic input sequence u[n] with period N .When this input is applied to the LTI system G for all time n, the resultingoutput sequence y[n] = Gu[n] is also periodic (period N), and thus has aDFS representation.Rewriting the sequences using their DFS representations, it follows that:

N−1∑

Y [k]ejk2πN

N−1∑

U [k]ejk2πN

Using linearity, this can be rewritten as:

∀n,1

N−1∑

Y [k] ejk2πN

N−1∑

U [k]ejk2πN

N−1∑

H(z = ejk 2π

N ) U [k] ejk2πN

where H(z) is the transfer function of system G.

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Outline

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N−1∑

Y [k] ejk2πN

N−1∑

H(z = ejk 2π

N ) U [k] ejk2πN

Comparing the coefficients of the complex exponential terms leads to, for allk ∈ Z, lead to the following relationship between the DFS coefficients:

Y [k] = H(ejk2πN )U [k].

This result shows that:

the DFS coefficients of the output sequence’s : Y [k] are related to the DFScoefficients of the input U [k] by the system’s transfer functionH(z), sampled

at z = ejk2πN .

Remark:Note that this is equivalent to sampling the discrete-time Fourier transformH(Ω) of the system’s impulse response h[n] at discrete frequencies Ω = k 2π

for k ∈ Z.

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DefinitionExample

Outline

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DefinitionExample

Consider the Fourier series representation of a periodic signal x with period N :

x[n] =1

N−1∑

X[k]ejk2πN

for all times n. In Lecture 4, we saw how to extend the Fourier transform todeal with sequences that are not absolutely summable. Using that result (andrecalling that a rigorous treatment would need an understanding of the theoryof distributions), we can find the Fourier transform of x as follows:

X(Ω) =

∞∑

n=−∞

N−1∑

X[k]ejk2πN

ne−jΩn

N−1∑

X[k]∞∑

n=−∞

ejn(k 2π

N−Ω)=

N−1∑

X[k]δ(Ω− k2π

1 Note that one would have to rigorously show that the order of summation can be swapped.

2 The DT Fourier transform of a periodic signal is therefore a finite sum of scaled and shifted Dirac delta

functions.

3 Note that the delta functions are located at the finite frequencies of the DFS, and scaled by the DFS

coefficients X[k].25 / 27

DefinitionExample

Outline

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DefinitionExample

Example : Consider the absolutely summable sequence x1[n], the sequencex2[n] which is periodic with N = 6, and the magnitudes of their DT Fouriertransforms.

We see that X2(Ω) is composed of Dirac functions located at the discretefrequencies of the DFS representation of x2[n]:

x2[n] =16

∑5k=0 X2[k]e

jk 2π6

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Signals and Systems - ETH Z€¦ · Discrete Fourier series representation of a periodic signal...

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