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Signals and Systems

Lecture 6: Fourier Analysis - Applied Concepts

Dr. Guillaume Ducard

Fall 2018

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

Institute for Dynamic Systems and Control

ETH Zurich, Switzerland

1 / 40

Outline

1The Discrete Fourier Transform

Overview

DFT: Definition

2The DFT: discussion

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

Problem statement and theorem

Proof

3Aliasing

Description

Example

4Annex: The fast Fourier transform

Problem statement

Example of a FFT calculation whenNis even

2 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Overview of Frequency Domain Analysis in Lectures 4 - 6 Tools for analysisof signals and systems in frequency domain: Lecture 4 - The DT Fourier transform (FT):For general, infinitely long and absolutely summable signals. ?Useful for theory and LTI system analysis. Lecture 5 - The discrete Fourier series (DFS):For infinitely long but periodic signals ?basis for the discrete Fourier transform. Lecture 6 - 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 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transformOverview

DFT: Definition

Outline

1The Discrete Fourier Transform

Overview

DFT: Definition

2The DFT: discussion

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

Problem statement and theorem

Proof

3Aliasing

Description

Example

4Annex: The fast Fourier transform

Problem statement

Example of a FFT calculation whenNis even

4 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transformOverview

DFT: Definition

Recap and overview

Recall from Lecture 5 : The discrete Fourier series (DFS) showed that an infinitely long and periodic signal can be represented as a finite sum of complex exponentials.

Synthesis:

x[n]=1NN-1? k=0X[k]ejk2π

Nnfor alln,

Analysis:

X[k]=N-1?

n=0x[n]e-jk2π

Nnfor allk.

In this lecture, we introduce:

the discrete-time discrete Fourier transform (DT DFT, or simply DFT). The DFT is very useful in practice: used forfinite lengthsequences, usually encountered in real-world experiments.

5 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transformOverview

DFT: Definition

Overview

Question:How to deal withfinite-lengthsignals with tools that are designed forinfinite-lengthsignals?

Main idea:

1{x[n]}be a sequence of finite lengthN

2and letxe[n]=x[nmodN],?n .

The sequence

xeis known as theperiodic extensionofx. x n? ?? ?? ?ex n? ?? ?? ? nn N

6 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transformOverview

DFT: Definition

Overview

Conclusion:theextended signalxeis periodicwith periodNand can therefore be represented by its discrete Fourier series (DFS):

Synthesisequation

xe[n]=1NN-1? k=0Xe[k]ejk2πNnfor all n,

Analysisequation

Xe[k]=N-1?

n=0xe[n]e-jk2πNnfor allk.

7 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transformOverview

DFT: Definition

Outline

1The Discrete Fourier Transform

Overview

DFT: Definition

2The DFT: discussion

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

Problem statement and theorem

Proof

3Aliasing

Description

Example

4Annex: The fast Fourier transform

Problem statement

Example of a FFT calculation whenNis even

8 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transformOverview

DFT: Definition

The Discrete Fourier Transform (DFT)

1The original sequencexis of finite lengthNand onlydefined for

n= 0,...,N-1

2The DFS representationof the signal"s periodic extension isvalid for all

n , and thusstill holdsover thisreduced interval.

The sequencexcan therefore be represented as

Synthesis:x[n]=1NN-1?k=0X[k]ejk2πNnforn= 0,...,N-1,

Analysis:X[k]=Xe[k]=N-1?

n=0x[n]e-jk2πNnfork= 0,...,N-1. The elements of sequence{X[k]}are called:DFT coefficientsof the signal. The DFT coefficientsof a finite-length signalare the DFS coefficientsof the signal"s periodic extension.

9 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

Outline

1The Discrete Fourier Transform

Overview

DFT: Definition

2The DFT: discussion

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

Problem statement and theorem

Proof

3Aliasing

Description

Example

4Annex: The fast Fourier transform

Problem statement

Example of a FFT calculation whenNis even

10 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

The DFT: discussion

We just saw the correspondence between the DFS and the DFT.?

It follows that the

DFT coefficients{X[k]}of a N-point sequence

willexist at finite frequenciesΩ =k2πNfork?[0,N-1].

Consider the complex exponential sequence :

{x[n]}={ejΩ0n}. We have shown in Lecture 4 that its Fourier transform is given by

X(Ω) = 2πδ(Ω-Ω0).

We can distinguish two cases:

1Ω0isan integer multiple of2πN

2Ω0is notan integer multiple of2πN11 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

DFT: discussion

Case 1:ifΩ0isan integer multiple of2πN

there exists ak0?[0,N-1], such that : k02π

N= Ω0.

?In this case, the DFT coefficientX[ k0]

1is located at the "location" of the delta function:δ(Ω-Ω0),

2and captures all of the signal"s power.

1The DT Fourier transform ofxis thenX(Ω) = 2πδ(Ω-Ω0).

2We choose to take a DFT of the firstNsamples ofx.

That is, letxN[n] =x[n]forn= 0,...,N-1, and let{XN[k]}be the

N-point DFT of{xN[n]}.12 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

The following figure shows the locations of the DFT coefficients forN= 12. Re(z) Im(z)

X12[0]

X12[1]

X12[2]

X12[3]

X12[4]

X12[5]

X12[6]

X12[7]

X12[8]

X12[9]

X12[10]

X12[11]

2π 12

X(Ω0)

Ω0

13 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

|X12[k]| 0π 6 1π 6 2π 6 3π 6 4π 6 5π 6 6π 6 7π 6 8π 6 9π 6

10π

6

11π

6 0 4 8 12

Conclusion:

Note thatX12[2]is at the same location as the delta function, since k 2π

N= Ω0forN= 12andk= 2.

Considering the magnitude of the DFT coefficients{X12[k]}, we see that the power of the sequence{x12[n]}is concentrated in the single DFT coefficientX12[2]

14 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

Outline

1The Discrete Fourier Transform

Overview

DFT: Definition

2The DFT: discussion

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

Problem statement and theorem

Proof

3Aliasing

Description

Example

4Annex: The fast Fourier transform

Problem statement

Example of a FFT calculation whenNis even

15 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

DFT: discussion

Case 2:ifΩ0is notan integer multiple of2πN

this can happen for two reasons:

1Nis not large enough;

2or the signal is simply not periodic (the case for almost alldiscrete-time, real-world signals).

?In this case, the signal"s power is spread over the DFT components.

This is illustrated in the following example.

16 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

Now letN= 10.

Note thatΩ0is not an integer multiple of2π

Nand thus no DFT coefficients

exist at the location of the delta function. Re(z) Im(z)

X10[0]

X10[1]

X10[2]X10[3]

X10[4]

X10[5]

X10[6]

X10[7]X10[8]

X10[9]

2π 10

X(Ω0)

Ω0

17 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

|X10[k]| 0π 5 1π 5 2π 5 3π 5 4π 5 5π 5 6π 5 7π 5 8π 5 9π 5 0 4 8 12 Conclusion:Since no DFT coefficient exists at the location of the delta function, the power of the signal is spread across the DFT coefficients.

18 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2π

NThe Effect of Causal Inputs

Outline

1The Discrete Fourier Transform

Overview

DFT: Definition

2The DFT: discussion

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2πNThe Effect of Causal Inputs

Problem statement and theorem

Proof

3Aliasing

Description

Example

4Annex: The fast Fourier transform

Problem statement

Example of a FFT calculation whenNis even

19 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2π

NThe Effect of Causal Inputs

Effect of causal inputs

Recall

Complex exponential sequences of the form{ejΩn}are eigenfunctions of a LTI system G:

G{ejΩn}=H(z=ejΩ){ejΩn},

whereH(z)is the transfer function of system G. Note that the input signal has bi-infinite length (u[n] =ejΩnfor all n?(-∞,∞)). We now investigate the effects of applying a causal input sequenceu, as would be the case in a real-world experiment.

20 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2π

NThe Effect of Causal Inputs

Effect of causal inputs

Letube defined as:

u[n] =? ejΩnn≥0

0n <0,

and lety=Gube the resulting output. In general, it is not true that y[n] =?

H(z=ejΩ)ejΩnn≥0

0n <0.

However, we have the following useful result:

Theorem

Let u[n] =? ejΩnn≥0

0n <0.

Let the LTI system G be stable

, and lety=Gu. Then y[n]→H(z=ejΩ)ejΩnasn→ ∞.

21 / 40

The Discrete Fourier Transform

The DFT: discussion

Aliasing

Annex: The fast Fourier transform

Case 1: ifΩ0is an integer multiple of2πNCase 2: ifΩ0is not an integer multiple of2π

NThe Effect of Causal Inputs

Proof:Let the LTI system G be stable, and let

u[n] =? ejΩnn≥0

0n <0v[n] =?

0n≥0

ejΩnn <0 y=Gu w=Gv. It is clear that{u[n] +v[n]}={ejΩn}. By linearity, we have {y[n] +w[n]}=G{u[n] +v[n]} =G{ejΩn} =H(z=ejΩ){ejΩn}. It follows thatw[n] =H(ejΩ)ejΩn-y[n]. Fromw=Gvand since w[n] =∞? k=-∞h[k]v[n-k] k=n+1h[k]v[n-k].

22 / 40

The Discrete Fourier Transform

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