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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

1 / 27

Outline

1The Discrete Fourier Series

Discrete Fourier series representation of a periodic signal

Properties of the discrete Fourier series

DFS coefficients of real signals

2Response to Complex Exponential Sequences

Complex exponential as input

DFS coefficients of inputs and outputs

3Relation between DFS and the DT Fourier Transform

Definition

Example

2 / 27

The Discrete Fourier Series

Response to Complex Exponential Sequences

Relation between DFS and the DT Fourier Transform

Discrete Fourier series representation of a periodic signal

Properties of the discrete Fourier series

DFS coefficients of real signals

Overview of Frequency Domain Analysis in Lectures 4 - 6 Tools for analysisof signals and systems in frequency domain: The DT Fourier transform (FT):For general, infinitely long and absolutely summable signals. ?Useful for theory and LTI system analysis. The discrete Fourier series (DFS):For infinitely long but periodic signals ?basis for the discrete Fourier transform. The discrete Fourier transform (DFT):For general, finite length signals. ?Used in practice with signals from experiments. Underlying these three concepts is the decomposition of signals into sums of sinusoids (or complex exponentials).3 / 27

The Discrete Fourier Series

Response to Complex Exponential Sequences

Relation between DFS and the DT Fourier Transform

Discrete Fourier series representation of a periodic signal

Properties of the discrete Fourier series

DFS coefficients of real signals

Outline

1The Discrete Fourier Series

Discrete Fourier series representation of a periodic signal

Properties of the discrete Fourier series

DFS coefficients of real signals

2Response to Complex Exponential Sequences

Complex exponential as input

DFS coefficients of inputs and outputs

3Relation between DFS and the DT Fourier Transform

Definition

Example

4 / 27

The Discrete Fourier Series

Response to Complex Exponential Sequences

Relation between DFS and the DT Fourier Transform

Discrete Fourier series representation of a periodic signal

Properties of the discrete Fourier series

DFS coefficients of real signals

Recall on periodic signals

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

Recall

A periodic sequencexwith periodNis such that

x[n+N] =x[n],?n

5 / 27

The Discrete Fourier Series

Response to Complex Exponential Sequences

Relation between DFS and the DT Fourier Transform

Discrete Fourier series representation of a periodic signal

Properties of the discrete Fourier series

DFS coefficients of real signals

Overview of the Discrete Fourier Series

Analysisequation (Discrete Fourier Series)

DFS coefficients are obtained as:

X[k]=N-1?

n=0x[n]e-jk2π Nn. Synthesisequation (Inverse Discrete Fourier Series) x[n] =1NN-1? k=0X[k]ejk2πNn

6 / 27

The Discrete Fourier Series

Response to Complex Exponential Sequences

Relation between DFS and the DT Fourier Transform

Discrete Fourier series representation of a periodic signal

Properties of the discrete Fourier series

DFS coefficients of real signals

Discrete Fourier series representation of a periodic signal The Discrete Fourier Series (DFS)is an alternative representation of a periodic sequencexwith periodN.

The periodic sequencexcan be represented

as a sumofNcomplex exponentialswithfrequenciesk2πN, where k= 0,1,...,N-1: x[n] =1 NN-1? k=0X[k]ejk2πNn(1) for all timesn, where

X[k]?Cis thekthDFS coefficient

corresponding to the complex exponential sequence{ejk2πNn}.

7 / 27

The Discrete Fourier Series

Response to Complex Exponential Sequences

Relation between DFS and the DT Fourier Transform

Discrete Fourier series representation of a periodic signal

Properties of the discrete Fourier series

DFS coefficients of real signals

Discrete Fourier series representation of a periodic signal

Remarks:

1Thefrequency2πNis called thefundamental frequencyand is

the lowest frequency componentin the signal. (We will later show that there is no loss of information in this representation.)

2We only needNcomplex exponentials to represent aDT

periodic signal with periodN Indeed, there are onlyNdistinct complex exponentials with frequencies that are integer multiples of 2π N: e j(k+N)2π

3graphical representation (shown during class).

8 / 27

The Discrete Fourier Series

Response to Complex Exponential Sequences

Relation between DFS and the DT Fourier Transform

Discrete Fourier series representation of a periodic signal

Properties of the discrete Fourier series

DFS coefficients of real signals

Discrete Fourier series coefficients

TheDFS coefficientsof Equation (1) are obtained from the periodic signalx(the role ofXandxare permuted with a minus sign in the exponential) as:

X[k]=N-1?

n=0x[n]e-jk2π Nn.

The DFS operator is denoted asF

s, where: X=Fsx andx=F-1sX. The pairs are usually denoted as{x[n]} ←→ {X[k]}.

9 / 27

The Discrete Fourier Series

Response to Complex Exponential Sequences

Relation between DFS and the DT Fourier Transform

Discrete Fourier series representation of a periodic signal

Properties of the discrete Fourier series

DFS coefficients of real signals

Discrete Fourier series coefficients

Note that, like the underlying sequencex,Xis periodic with periodN

Proof:

X[k+N] =N-1?

n=0x[n]e-j(k+N)2π Nn N-1? n=0x[n]e-jk2π

Nne-j2πn????

=1?n=X[k].

Conclusions

When working with the DFS, it is therefore common practice to only consider one period of the sequence{X[k]}, that is:only the

NDFS coefficients as

X[k]fork= 0,1,...,N-1.

10 / 27

The Discrete Fourier Series

Response to Complex Exponential Sequences

Relation between DFS and the DT Fourier Transform

Discrete Fourier series representation of a periodic signal

Properties of the discrete Fourier series

DFS coefficients of real signals

Proof that the DFS operator in invertible - Part I The following identity highlights the orthogonality of complex exponentials: 1 NN-1? n=0e j(r-k)2π Nn=?

1forr-k=mN, m?Z

0otherwise.

Proof:

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

e jmN2π

Nn=ej2πmn= 1for allm,n

1 NN-1? n=0e j(r-k)2π

Nn=1NN-1?

n=01 =1NN= 1.

Case 2 :r-k?=mN. Definel:=r-k. We then have

1 NN-1? n=0e jl2π

Nn=1N1-ejl2π

NN

1-ejl2πN,

the above equation is a geometric series andejl2π

N?= 1sincel?=mN. For

l?=mN, we therefore have1

N1-ej2πl1-ejl2πN= 0.

11 / 27

The Discrete Fourier Series

Response to Complex Exponential Sequences

Relation between DFS and the DT Fourier Transform

Discrete Fourier series representation of a periodic signal

Properties of the discrete Fourier series

DFS coefficients of real signals

Proof that the DFS operator in invertible - Part II In order to prove thatF-1sis the inverse transform ofFs, we need to show:

1FsF-1s=I

2andF-1sFs=I, whereIis the identity operator.

We now show thatFsF-1s=I:

FsF-1s{X[k]}=?

N-1? n=0? 1 NN-1? r=0X[r]ejr2π

Nn?e-jk2πNn

N-1? r=0X[r] (1 N N-1? n=0 ej(r-k)2πNnquotesdbs_dbs17.pdfusesText_23