Chapter 4 Fourier Transform of Discrete-Time Signals 2 nd lecture Mon. June 17, 2013
Signals and Systems - ETH Z€¦ · Discrete Fourier series representation of a periodic signal...
Transcript of 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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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
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
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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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 the Discrete Fourier Series
Analysis equation (Discrete Fourier Series)
DFS coefficients are obtained as:
X[k] =
N−1∑
n=0
x[n]e−jk 2πN
n.
Synthesis equation (Inverse Discrete Fourier Series)
x[n] =1
N
N−1∑
k=0
X[k]ejk2πN
n
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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
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
N−1∑
k=0
X[k]ejk2πN
n (1)
for all times n, where X[k] ∈ C is the kth DFS coefficient
corresponding to the complex exponential sequence ejk2πN
n.
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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
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
N:
ej(k+N) 2πN
n = ejk2πN
nejN2πN
n = ejk2πN
nej2πn = ejk2πN
n.
3 graphical representation (shown during class).
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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
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∑
n=0
x[n]e−jk 2πN
n.
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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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
Discrete Fourier series coefficients
Note that, like the underlying sequence x, X is periodic withperiod N :Proof:
X[k +N ] =
N−1∑
n=0
x[n]e−j(k+N) 2πN
n
=
N−1∑
n=0
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
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
Proof that the DFS operator in invertible - Part I
The following identity highlights the orthogonality of complex exponentials:
1
N
N−1∑
n=0
ej(r−k) 2π
Nn =
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
∴1
N
N−1∑
n=0
ej(r−k) 2π
Nn =
1
N
N−1∑
n=0
1 =1
NN= 1.
Case 2 : r − k 6= mN . Define l := r − k. We then have
1
N
N−1∑
n=0
ejl 2π
Nn =
1
N
1− ejl2πN
N
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
N
= 0.11 / 27
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
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∑
n=0
(
1
N
N−1∑
r=0
X[r]ejr2πN
n
)
e−jk 2π
Nn
=
N−1∑
r=0
X[r]
1
N
N−1∑
n=0
ej(r−k) 2π
Nn
︸ ︷︷ ︸
.
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∑
r=0
X[r]
(
1
N
N−1∑
n=0
ej(r−k) 2π
Nn
)
= X[k mod N ] = X[k].
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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
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
N
∑
k=〈N〉
X[k]ejk2πN
n and X[k] =∑
n=〈N〉
x[n]e−jk 2πN
n.
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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
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
Properties of the discrete Fourier series
Linearity
a1x1[n]+ a2x2[n] ←→ a1X1[k]+ a2X2[k]
Parseval’s theorem (recall: period is N)
N−1∑
n=0
|x[n]|2 =1
N
N−1∑
k=0
|X[k]|2
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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
Proof of Parseval’s theorem
1
N
N−1∑
k=0
|X[k]|2 =?
=1
N
N−1∑
k=0
X∗[k]X[k] (where ∗ denotes the complex conjugate)
=1
N
N−1∑
k=0
(N−1∑
n=0
x∗[n]ejk
2πN
n
)
X[k]
=1
N
N−1∑
k=0
N−1∑
n=0
x∗[n]X[k]ejk
2πN
n
=N−1∑
n=0
x∗[n]
1
N
N−1∑
k=0
X[k]ejk2πN
n
︸ ︷︷ ︸
x[n]
=
N−1∑
n=0
x∗[n]x[n] =
N−1∑
n=0
|x[n]|2
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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
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
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∑
n=0
x[n]e−jk 2πN
n.
Letting k = N − γ, where γ is an integer, leads to
X[N − γ] =N−1∑
n=0
x[n]e−j(N−γ) 2πN
n
=
N−1∑
n=0
x[n] e−j2πn
︸ ︷︷ ︸=1∀n
ejγ 2π
Nn =
N−1∑
n=0
x[n]ejγ2πN
n =
N−1∑
n=0
x∗[n]ejγ
2πN
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
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
DFS coefficients of real signals
Example
Consider the periodic signal :
x = . . . ,1
2↑
, 2,1
2, 1, . . . ,
Questions:
1 What is the period of such signal ?
2 Compute the Discrete Fourier Series coefficients.
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The Discrete Fourier SeriesResponse to Complex Exponential Sequences
Relation between DFS and the DT Fourier Transform
Complex exponential as inputDFS coefficients of inputs and outputs
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
Complex exponential as inputDFS coefficients of inputs and outputs
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:
1
N
N−1∑
k=0
Y [k]ejk2πN
n
= G
1
N
N−1∑
k=0
U [k]ejk2πN
n
.
Using linearity, this can be rewritten as:
∀n,1
N
N−1∑
k=0
Y [k] ejk2πN
n =1
N
N−1∑
k=0
G(
U [k]ejk2πN
n)
=1
N
N−1∑
k=0
H(z = ejk 2π
N ) U [k] ejk2πN
n,
where H(z) is the transfer function of system G.
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The Discrete Fourier SeriesResponse to Complex Exponential Sequences
Relation between DFS and the DT Fourier Transform
Complex exponential as inputDFS coefficients of inputs and outputs
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
Complex exponential as inputDFS coefficients of inputs and outputs
1
N
N−1∑
k=0
Y [k] ejk2πN
n =1
N
N−1∑
k=0
H(z = ejk 2π
N ) U [k] ejk2πN
n.
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
N
for k ∈ Z.
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The Discrete Fourier SeriesResponse to Complex Exponential Sequences
Relation between DFS and the DT Fourier Transform
DefinitionExample
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
DefinitionExample
Relation between DFS and the DT Fourier Transform
Consider the Fourier series representation of a periodic signal x with period N :
x[n] =1
N
N−1∑
k=0
X[k]ejk2πN
n
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=−∞
1
N
N−1∑
k=0
X[k]ejk2πN
ne−jΩn
=1
N
N−1∑
k=0
X[k]∞∑
n=−∞
ejn(k 2π
N−Ω)=
2π
N
N−1∑
k=0
X[k]δ(Ω− k2π
N).
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
The Discrete Fourier SeriesResponse to Complex Exponential Sequences
Relation between DFS and the DT Fourier Transform
DefinitionExample
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
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
n
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