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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)2πNn From above equation, the term in underbrace is equal to1forr=kmodN and0otherwise. Thus: F sF-1s{X[k]}=? N-1? r=0X[r]? 1 NN-1? n=0e j(r-k)2π Nn?? ={X[kmodN]}={X[k]}.
12 / 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
In a similar way, it can also be shown thatF-1sFs=I.
Remark:
As bothxandXare periodic with periodN, we can sum over any
Nconsecutive values (denoted as?N?):
x[n] =1 N? k=?N?X[k]ejk2π
NnandX[k] =?
n=?N?x[n]e-jk2πNn.
13 / 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
14 / 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
Properties of the discrete Fourier series
Linearity
a1{x1[n]}+a2{x2[n]} ←→a1{X1[k]}+a2{X2[k]}
Parseval"s theorem (recall: period isN)
N-1? n=0|x[n]|2=1 NN-1? k=0|X[k]|2
15 / 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 of Parseval"s theorem
1
NN-1?k=0|X[k]|2=?=1
NN-1?k=0X?[k]X[k](where?denotes the complex conjugate) 1
NN-1?k=0
?N-1? n=0x ?[n]ejk2π
Nn?X[k]
1
NN-1?k=0N-1?
n=0x ?[n]X[k]ejk2π Nn N-1? n=0x ?[n]1
NN-1?k=0X[k]ejk2π
Nn x[n] N-1? n=0x ?[n]x[n] =N-1? n=0|x[n]|2
16 / 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
17 / 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
DFS coefficients of real signals:x[n]?Rfor alln
1So far:x[n]?Cfor alln.
2Now we consider:x[n]?Rfor alln; (most often case in practice)
Recall:DFS coefficients of a periodicsignalx:X[k] =N-1? n=0x[n]e-jk2πNn.
Lettingk=N-γ, whereγis an integer, leads to
X[N-γ]=N-1?
n=0x[n]e-j(N-γ)2πNn N-1? n=0x[n]e-j2πn? =1?ne jγ2π
Nn=N-1?
n=0x[n]ejγ2πNn=N-1? n=0x ?[n]ejγ2πNn=X?[γ],
We used that for arealsignalx,x[n] =x?[n].
Conclusions:
For areal signalx, we have thatX[N-k] =X?[k].
Takeγ= 0→X[N] =X?[0],γ=N→X[0] =X?[N].
By periodicityX[0] =X[N]. ThusX[0] =X?[0].
For areal signal,X[0]is thereforealways real.
quotesdbs_dbs17.pdfusesText_23
[PDF] Chapter 3 Fourier Series Representation of Period Signals