Fourier analysis - CAL UniBg · Scandella Matteo - Dynamical System Identification course 23 Uniqueness Given a periodic signal its Fourier coefficients are unique and it’s the

Fourier analysisDynamic System Identification – Part 1, Lecture 6

Signal typesA signal is a real function of time

2Scandella Matteo - Dynamical System Identification course

𝑠 ∶ ℝ → ℝ 𝑠 ∶ ℤ → ℝ

Continuous signal Discrete signal

Sampling

Reconstruction

The reconstruction is possible only in certain cases.

Continuous signalA signal is a real function in real variable:

𝑠 ∶ ℝ → ℝ


Example: Step


𝑠𝑡𝑒𝑝 𝑡 = ቊ1, 𝑡 ≥ 00, 𝑡 < 0

Example: Ramp


𝑟𝑎𝑚𝑝 𝑡 = ቊ𝑡, 𝑡 ≥ 00, 𝑡 < 0

Example: Dirac delta


𝛿 𝑡 = ቊ∞, 𝑡 = 00, 𝑡 ≠ 0

න−∞

+∞

𝛿 𝑡 𝑑𝑡 = 1with

Periodic signal


A signal 𝑠 is called periodic if and only if ∃𝑇 ∈ ℝ+, called period, such that:

∀𝑘 ∈ ℤ, 𝑠 𝑡 = 𝑠 𝑡 + 𝑘 ⋅ 𝑇0

𝑇

Example: Sine wave


𝑠 𝑡 = 𝐴 ⋅ sin 2 ⋅ 𝜋 ⋅𝑡

𝑇+ 𝜑

𝐴: Amplitude

𝑇: Period

𝜑: Phase

sin−1 𝜑

Remark on periodic signal


A periodic signal is defined by the behavior

inside one period:

𝑠𝑝 ∶ (0, 𝑇] → ℝ

Because:

𝑠 𝑡 =

𝑘=−∞

𝑘=∞

𝑠𝑝 𝑡 + 𝑘𝑇

It’s also possible to note that every section of the

function domain with length 𝑇 can be used. For

example:

𝑠𝑝2 ∶ −𝑇

2,𝑇

2→ ℝ

Remark on the period • The period value 𝑇 is not uniquely define

• If a signal is periodic with period 𝑇 then it’s also periodic with period 𝑘 ⋅ 𝑇 where 𝑘 is a positive natural number.

• The smallest possible period (greater than zero) will be denoted with 𝑇0 and 𝑇𝑘 = 𝑘 ⋅ 𝑇0


𝑇0𝑇2 = 2𝑇0

𝑇3 = 3𝑇0

Frequency

Given a periodic signal 𝑠 𝑡 with period 𝑇0 we define its frequency 𝑓0 as the number of time that the function repeats itself in one unit of time:

𝑓0 =1

𝑇0The frequency is measured in:

𝐻𝑧 =1

𝑠


𝑓0 ↑ 𝑇0 ↓

Fourier seriesThe Fourier series is a way to approximate a periodic signal as a sum of sine waves.

Consider the periodic signal 𝑠 𝑡 with period 𝑇0:

𝑆𝑛 𝑠 𝑡 =𝑎02+

𝑘=1

𝑛

𝑎𝑘 ⋅ cos2𝜋𝑘𝑡

𝑇0+ 𝑏𝑘 ⋅ sin

2𝜋𝑘𝑡

𝑇0


𝑎𝑘 =2

𝑇0⋅ න

0

𝑇0

𝑠 𝑡 ⋅ cos2𝜋𝑘𝑡

𝑇0𝑑𝑡 , 𝑘 = 0,1,⋯ , 𝑛

𝑏𝑘 =2

𝑇0⋅ න

0

𝑇0

𝑠 𝑡 ⋅ sin2𝜋𝑘𝑡

𝑇0𝑑𝑡 , 𝑘 = 1,⋯ , 𝑛

lim𝑛→∞

𝑆𝑛 𝑠 𝑡 = 𝑠 𝑡

With some assumptions

Example: square wave


𝑠𝑝 𝑥 =𝐴 0 ≤ 𝑡 <

𝑇02

−𝐴𝑇02≤ 𝑡 < 𝑇

𝑎𝑘 = 0

𝑏𝑘 = ቐ0 if 𝑘 is even4𝐴

𝜋𝑘if 𝑘 is 𝑜𝑑𝑑

𝑆1 𝑠 𝑡

𝑠 𝑡



𝑠𝑝 𝑥 =𝐴 0 ≤ 𝑡 <

𝑇02

−𝐴𝑇02≤ 𝑡 < 𝑇

𝑎𝑘 = 0



𝑆3 𝑠 𝑡

𝑠 𝑡

3-th sine wave



𝑠𝑝 𝑥 =𝐴 0 ≤ 𝑡 <

𝑇02

−𝐴𝑇02≤ 𝑡 < 𝑇

𝑎𝑘 = 0



𝑆5 𝑠 𝑡

𝑠 𝑡

5-th sine wave



𝑠𝑝 𝑥 =𝐴 0 ≤ 𝑡 <

𝑇02

−𝐴𝑇02≤ 𝑡 < 𝑇

𝑎𝑘 = 0



𝑆23 𝑠 𝑡

𝑠 𝑡

23-th sine

wave

Harmonics• The Fourier series decomposes a periodic signal in

a sum of sine waves called harmonics.

• The 𝑘-th harmonics has the period:

• The first harmonics are called fundamental harmonic and its period is equal to 𝑇0

• All the other harmonics have frequency multiple of 𝑇0


𝑇𝑘 =𝑇0𝑘

𝑓𝑘 =1

𝑇𝑘=

𝑘

𝑇0= 𝑘 ⋅ 𝑓0

Angular velocityGiven a sine wave:

𝑠 𝑡 = 𝐴 ⋅ sin 2 ⋅ 𝜋 ⋅𝑡

𝑇0+ 𝜑

The angular velocity 𝜔0 is defined by number of radiants of the sin argument that change in the time unit.

The angle is:

𝜃 = 2 ⋅ 𝜋 ⋅𝑡

𝑇0+ 𝜑

And the velocity

𝜔0 =𝑑𝜃

𝑑𝑡=2 ⋅ 𝜋

𝑇0= 2 ⋅ 𝜋 ⋅ 𝑓0

The angular velocity of the harmonics of a generic periodic signal are the angular velocity of the generic signal.


𝜔𝑘 =2𝜋

𝑇𝑘= 2𝜋𝑓0

Constant componentRecalling the Fourier series:


𝑘=1

𝑛



2𝜋𝑘𝑡

𝑇0

The term 𝑎0

2is called offset of the signal or constant component.


𝑎0 = 0 𝑎0 ≠ 0

𝑎02

Complex formRecalling the Euler’s formula:

𝑒𝑗𝑥 = cos 𝑥 + 𝑗 ⋅ sin 𝑥

It’s possible to write:


𝑘=1

𝑛



2𝜋𝑘𝑡

𝑇0

=

𝑘=−𝑛

𝑛

𝑐𝑘 ⋅ 𝑒𝑗⋅2𝜋𝑘𝑡𝑇0

where:

𝑐𝑘 =1

𝑇0⋅ න

0

𝑇0

𝑠 𝑡 ⋅ 𝑒−𝑗⋅

2𝜋𝑘𝑡𝑇0 𝑑𝑡 , 𝑘 = −𝑛,⋯ ,−1,0,1,⋯ , 𝑛

The coefficients 𝑐 are a complex sequence 𝑐: ℤ → ℂ that define the amplitude of the harmonics that compose the signal.

In the complex form, the constant component is 𝑐0 =𝑎0

2∈ ℝ.


𝑘-th complex harmonic

Amplitude of the 𝑘-th complex harmonic

Properties of 𝑐𝑘


𝑠 is even 𝑠 𝑡 = 𝑠 −𝑡 𝑐−𝑘 = 𝑐𝑘 𝑐 is even

𝑠 is odd 𝑠 𝑡 = −𝑠 −𝑡 𝑐−𝑘 = −𝑐𝑘 𝑐 is odd

𝑠 𝑡 ∈ ℝ 𝑐−𝑘 = 𝑐𝑘∗

Re 𝑐 is even

Im 𝑐 is odd

Re 𝑐−𝑘 = Re 𝑐𝑘

Im 𝑐−𝑘 = −Im 𝑐𝑘

For us this property is always valid, we won’t look into

complex signals

𝑐−𝑘 = 𝑐𝑘

∠𝑐−𝑘 = −∠𝑐𝑘

𝑐𝑘 is even

∠𝑐𝑘 is odd

LinearityGiven two signal and their Fourier coefficient:

𝑠𝐴 𝑡 ⇆ 𝑐𝐴,𝑘𝑠𝐵 𝑡 ⇆ 𝑐𝐵,𝑘

the signal𝑠 𝑡 = 𝛼𝑠𝐴 𝑡 + 𝛽𝑠𝐵 𝑡

has coefficients:

𝑐𝑘 =2

𝑇0⋅ න

0

𝑇0


2𝜋𝑘𝑡𝑇0 𝑑𝑡

=2

𝑇0⋅ න

0

𝑇0

𝛼𝑠𝐴 𝑡 + 𝛽𝑠𝐵 𝑡 ⋅ 𝑒−𝑗⋅


= 𝛼2

𝑇0⋅ න

0

𝑇0

𝑠𝐴 𝑡 ⋅ 𝑒−𝑗⋅

2𝜋𝑘𝑡𝑇0 𝑑𝑡 + 𝛽

2

𝑇0⋅ න

0

𝑇0

𝑠𝐵 𝑡 ⋅ 𝑒−𝑗⋅


= 𝛼𝑐𝐴,𝑘 + 𝛽𝑐𝐵,𝑘



Uniqueness

Given a periodic signal 𝑠 𝑡 its Fourier coefficients are unique

and it’s the only signal with such Fourier coefficients.

𝑠 𝑡 ⇄ 𝑐𝑘

𝒄𝟎 is the mean of the signal

Given a periodic signal 𝑠 𝑡 its first Fourier coefficient is:

𝑐0 =1

𝑇0⋅ න

0

𝑇0


2𝜋⋅0⋅𝑡𝑇0 𝑑𝑡 =

1

𝑇0⋅ න

0

𝑇0

𝑠 𝑡 ⋅ 𝑑𝑡

𝑐0 ∈ ℝ

Interpretation for real signals


𝑆𝑛 𝑠 𝑡 =

𝑘=−𝑛

𝑛


=

𝑘=−𝑛

−1

𝑐𝑘 ⋅ 𝑒𝑗⋅2𝜋𝑘𝑡𝑇0 + 𝑐0𝑒

𝑗⋅2𝜋⋅0⋅𝑡𝑇0 +

𝑘=1

𝑛


=

𝑘=−𝑛

−1

𝑐−𝑘 ⋅ 𝑒𝑗⋅2𝜋𝑘𝑡𝑇0 + 𝑐0 +

𝑘=1

𝑛


=

𝑥=1

𝑛

𝑐𝑥 ⋅ 𝑒−𝑗⋅

2𝜋𝑥𝑡𝑇0 + 𝑐0 +

𝑘=1

𝑛


= 𝑐0 +

𝑘=1

𝑛

𝜌𝑘 ⋅ 𝑒−𝑗⋅𝜙𝑘 ⋅ 𝑒

−𝑗⋅2𝜋𝑥𝑡𝑇0 + 𝜌𝑘 ⋅ 𝑒

𝑗⋅𝜙𝑘 ⋅ 𝑒𝑗⋅2𝜋𝑘𝑡𝑇0

= 𝑐0 +

𝑘=1

𝑛

𝜌𝑘 ⋅ 𝑒−𝑗⋅

2𝜋𝑘𝑡𝑇0

+𝜙𝑘 + 𝑒𝑗⋅

2𝜋𝑥𝑡𝑇0

+𝜙𝑘

= 𝜌0 +

𝑘=1

𝑛

2𝜌𝑘 ⋅ cos2𝜋𝑘𝑡

𝑇0+ 𝜙𝑘

𝑐𝑘 = 𝜌𝑘 ⋅ 𝑒𝑗⋅𝜙𝑘

For a real signal, the couple

of the harmonics 𝑘 and −𝑘correspond to a cosine

Since 𝑐0 is real:

𝑐0 = 𝜌0

Spectral componentsThe succession 𝑐𝑘 contains information on how much the signal varies in the period.


The signal is varying slowly

Higher low

frequency

cosines

Spectral componentsThe succession 𝑐𝑘 contains information on how much the signal varies in the period.


The signal is varying quickly

Higher high

frequency

cosines



𝑠𝑝 𝑥 =𝐴 0 ≤ 𝑡 <

𝑇02

−𝐴𝑇02≤ 𝑡 < 𝑇

𝑐𝑘 = ቐ0 If 𝑘 is even

−𝑗 ⋅2𝐴

𝜋𝑘If 𝑘 is odd

Example: triangle wave


𝑠𝑝 𝑥 =2 ⋅ 𝐴

𝑇0⋅ 𝑡 𝑐𝑘 =

0 If 𝑘 ≠ 0 and it is even

−2𝐴

𝜋2𝑘2If 𝑘 is odd

𝑇0𝐴

2if 𝑘 = 0

Fourier transformGiven the non-periodical signal 𝑠 its Fourier transform is:

𝑆 𝑓 = ℱ 𝑠 𝑓 = න

−∞

+∞

𝑠 𝑡 ⋅ 𝑒−𝑗⋅2𝜋𝑓𝑡𝑑𝑡

The Fourier transform can be seen as a generalization of the Fourier series for non-periodical signal.

• Since 𝑇0 → ∞ and 𝑓0 → 0, the harmonics have an infinitesimal distance between them and therefore the coefficient succession 𝑐𝑘 becomes a function 𝑆 𝑓

• Since 𝑇0 → ∞ the integration over the period becomes the integration over the complete domain


Fourier transform properties


𝑠 is even 𝑠 𝑡 = 𝑠 −𝑡 𝑆 𝑓 = 𝑆 −𝑓 𝑆 is even

𝑠 is odd 𝑠 𝑡 = −𝑠 −𝑡 𝑆 𝑓 = −𝑆 −𝑓 𝑆 is odd

𝑠 𝑡 ∈ ℝ

𝑆 −𝑓 = 𝑆 𝑓 ∗

Re 𝑆 𝑓 is even

Im 𝑆 𝑓 is odd

Re 𝑆 −𝑓 = Re 𝑆 𝑓

Im 𝑆 −𝑓 = −Im 𝑆 𝑓

For us this property is always valid, we won’t look into

complex signals

𝑆 −𝑓 = 𝑆 𝑓

∠𝑆 −𝑓 = −∠𝑆 𝑓

𝑆 𝑓 is even

∠𝑆 𝑓 is odd

LinearityGiven two signal and their Fourier coefficient:

𝑠𝐴 𝑡 ⇆ 𝑆𝐴 𝑓𝑠𝐵 𝑡 ⇆ 𝑆𝐵 𝑓

the signal𝑠 𝑡 = 𝛼𝑠𝐴 𝑡 + 𝛽𝑠𝐵 𝑡

has coefficients:

ℱ 𝑠 𝑓 = න0

𝑇0

𝑠 𝑡 ⋅ 𝑒−𝑗⋅2𝜋𝑓𝑡𝑑𝑡

= න0

𝑇0

𝛼𝑠𝐴 𝑡 + 𝛽𝑠𝐵 𝑡 ⋅ 𝑒−𝑗⋅2𝜋𝑓𝑡𝑑𝑡

= 𝛼න0

𝑇0

𝑠𝐴 𝑡 ⋅ 𝑒−𝑗⋅2𝜋𝑓𝑡𝑑𝑡 + 𝛽න0

𝑇0

𝑠𝐵 𝑡 ⋅ 𝑒−𝑗⋅2𝜋𝑓𝑡𝑑𝑡

= 𝛼𝑆𝐴 𝑓 + 𝛽𝑆𝐵 𝑓


Fourier anti-transformGiven a signal 𝑠 𝑡 and its transformation 𝑆 𝑓 =ℱ 𝑠 𝑓 we have:

𝑠 𝑡 = න

−∞

+∞

𝑆 𝑓 ⋅ 𝑒𝑗⋅2𝜋𝑓𝑡𝑑𝑓


Given a signal 𝑠 𝑡 its Fourier transform is unique and it’s the only signal with such Fourier transform.

𝑠 𝑡 ⇄ 𝑆 𝑓

Spectra of a signal• The graph of the Fourier transform is called

spectra of the signal.

• The information contained in the spectra is similar to the one shown with Fourier coefficient for periodic signals.

• Since we are working with real signal we can show only the positive part of the spectra.


Example: Dirac delta


𝛿 𝑡 = ቊ∞, 𝑡 = 00, 𝑡 ≠ 0

න−∞

+∞

𝑓 𝑡 𝛿 𝑡 𝑑𝑡 = 𝑓 0

ℱ 𝛿 𝑓 = න

−∞

+∞

𝛿 𝑡 ⋅ 𝑒−𝑗⋅2𝜋𝑓𝑡𝑑𝑡

= 𝑒−𝑗⋅2𝜋𝑓0

= 1

Example: step


ℱ 𝑠𝑡𝑒𝑝 𝑓 = න

−∞

+∞

𝑠𝑡𝑒𝑝 𝑡 ⋅ 𝑒−𝑗⋅2𝜋𝑓𝑡𝑑𝑡

= න

−∞

0

0𝑑𝑡 + න

0

+∞

𝑒−𝑗⋅2𝜋𝑓𝑡𝑑𝑡

=𝑒−𝑗⋅2𝜋𝑓𝑡

−𝑗 ⋅ 2𝜋𝑓0

+∞

= 0 −1

−𝑗 ⋅ 2𝜋𝑓=

1

𝑗 ⋅ 2𝜋𝑓

𝑠𝑡𝑒𝑝 𝑡 = ቊ1, 𝑡 ≥ 00, 𝑡 < 0

Periodic signal transformGiven a periodic signal 𝑠𝑝 𝑡 and its Fourier coefficient 𝑐𝑘, we have:

ℱ 𝑠𝑝 𝑓 =

𝑘=−∞

+∞

𝑐𝑘 ⋅ 𝛿 𝑓 − 𝑓0 ⋅ 𝑘

Therefore the Fourier transform of a periodic signal is a weighted sum of Dirac deltas.


Example: Sine wave


𝑠 𝑡 = 𝐴 ⋅ cos 2 ⋅ 𝜋 ⋅𝑡

𝑇0

𝑐𝑘 = ቐ

𝐴

2𝑘 = 1

0 𝑘 ≠ 1

ℱ 𝑠𝑝 𝑓 =

𝑘=−∞

+∞

𝑐𝑘 ⋅ 𝛿 𝑓 − 𝑓0 ⋅ 𝑘

=𝐴

2𝛿 𝑓 − 𝑓0 +

𝐴

2𝛿 𝑓 + 𝑓0

Band limited signalsA band limited signal is a signal 𝑠 𝑡 such that ∃𝑓𝑚𝑎𝑥 ∈ ℝ such that:

ℱ 𝑠𝑝 𝑓 = 0, ∀𝑓 > 𝑓𝑚𝑎𝑥


Examples:

• The sine wave is a band limited signal

• The square wave is not a band limited signal

Fourier analysis - CAL UniBg · Scandella Matteo - Dynamical System Identification course 23 Uniqueness Given a periodic signal its Fourier coefficients are unique and it’s the

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