6.003 signal processingย ยท ๐ดโ, in terms of f[k]? previously(lecture 2b) let x[k] represent the...
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6.003 Signal Processing
Week 3, Lecture A:CTFS properties
6.003 Fall 2020
Continuous Time Fourier Series
Synthesis:
Analysis:
Today: useful properties of CTFS
Properties of CTFS: Linearity
โข Consider ๐ฆ ๐ก = ๐ด๐ฅ1 ๐ก + ๐ต๐ฅ2 ๐ก , where ๐ฅ1 ๐ก and ๐ฅ2 ๐ก are periodic in T. What are the CTFS coefficients ๐[๐], in terms of ๐1[๐]and ๐2[๐] ?
6.003 Fall 2020
First, ๐ฆ ๐ก must also be periodic in T
Properties of CTFS: Time flip
โข Consider ๐ฆ ๐ก = ๐ฅ โ๐ก , where ๐ฅ ๐ก is periodic in T. What are the CTFS coefficients ๐[๐], in terms of ๐[๐]?
6.003 Fall 2020
First, ๐ฆ ๐ก must also be periodic in T
๐ฅ ๐ก =
๐=โโ
โ
๐[๐]๐๐2๐๐๐ก๐ ๐ฆ ๐ก = ๐ฅ(โ๐ก) =
๐=โโ
โ
๐[๐]๐๐2๐๐(โ๐ก)
๐
Let ๐ = โ๐
๐ฆ ๐ก = ๐ฅ(โ๐ก) =
๐=โโ
โ
๐[โ๐]๐๐2๐๐๐ก๐
๐ฆ ๐ก =
๐=โโ
โ
๐[๐]๐๐2๐๐๐ก๐
Since we know
๐[๐] = ๐[โ๐]
If ๐ฆ ๐ก = ๐ฅ โ๐ก , ๐[๐] = ๐[โ๐]
Symmetric and Antisymmetric Parts
โข If ๐ ๐ก = ๐๐ ๐ก + ๐๐ด(๐ก) is a real-valued signal and periodic in time with fundamental period T, what are the Fourier coefficients of ๐๐ โ and ๐๐ด โ , in terms of F[k]?
6.003 Fall 2020
๐ ๐ก = ๐๐ ๐ก + ๐๐ด(๐ก)
Any arbitrary signal ๐ ๐ก can be written into two parts:
๐๐ ๐ก is symmetric about ๐ก = 0, ๐๐ ๐๐ ๐ก = ๐๐ โ๐ก for all t.
๐๐ด ๐ก is antisymmetric about ๐ก = 0, ๐๐ ๐๐ด ๐ก = โ๐๐ โ๐ก for all t.
๐๐ ๐ก =๐ ๐ก + ๐(โ๐ก)
2๐๐ด ๐ก =
๐ ๐ก โ ๐(โ๐ก)
2
Real-valued periodic signal
6.003 Fall 2020
๐น[๐] =1
๐เถฑ๐
๐(๐ก)๐โ๐2๐๐๐ก๐ ๐๐ก
If ๐ ๐ก is real valued periodic signal:
๐น[โ๐] =1
๐เถฑ๐
๐(๐ก)๐๐2๐๐๐ก๐ ๐๐ก
๐นโ[โ๐] =1
๐เถฑ๐
๐(๐ก)๐โ๐2๐๐๐ก๐ ๐๐ก
If ๐ ๐ก is real valued periodic signal, ๐น ๐ = ๐นโ[โ๐]
= ๐น[๐]
Symmetric and Antisymmetric Parts
If ๐ ๐ก is real valued periodic signal, ๐น ๐ = ๐นโ[โ๐]
๐๐ ๐ก =๐ ๐ก + ๐(โ๐ก)
2
๐ฟ๐๐๐๐๐๐๐ก๐ฆ๐น๐ ๐ =
๐น ๐ + ๐น โ๐
2=๐น ๐ + ๐นโ ๐
2=2๐ ๐(๐น ๐ )
2= ๐ ๐(๐น ๐ )
๐๐ด ๐ก =๐ ๐ก โ ๐(โ๐ก)
2
๐ฟ๐๐๐๐๐๐๐ก๐ฆ๐น๐ด ๐ =
๐น ๐ โ ๐น โ๐
2=๐น ๐ โ ๐นโ ๐
2=2๐ โ ๐ผ๐(๐น ๐ )
2= ๐ โ ๐ผ๐(๐น ๐ )
The real part of F[k] comes from the symmetric part of the signal, the imaginary part of F k comes from the antisymmetric part of the signal
โข If ๐ ๐ก = ๐๐ ๐ก + ๐๐ด(๐ก) is a real valued signal and periodic in time with fundamental period T, what are the Fourier coefficients of ๐๐ โ and ๐๐ด โ , in terms of F[k]?
Previously(lecture 2B)
Let X[k] represent the Fourier series coefficients of the following signal:
T=1, ๐0 =2๐
๐= 2๐
๐[๐] = แ
0, ๐ ๐๐ ๐๐ฃ๐๐1
๐๐๐, ๐ ๐๐ ๐๐๐
๐ฅ ๐ก is antisymmetric around t=0, thus its Fourier Series coefficients are purely imaginary
Symmetric and Antisymmetric Parts in Trig form CTFS
๐๐ ๐ก =๐ ๐ก + ๐(โ๐ก)
2๐๐ด ๐ก =
๐ ๐ก โ ๐(โ๐ก)
2
The symmetric part shows up in the ๐๐ coefficients, and the antisymmetric part shows up in the ๐๐ coefficients.
โข ๐๐โs (cosines) alone can only represent symmetric functions.
โข ๐๐ โs (sines) alone can only represent antisymmetric functions
sin(๐ฅ) cos(๐ฅ)
Previously(Lecture 2A)โข Find the Fourier series coefficients for the following square wave:
If without ๐0 =1
2โDCโ part, ๐ ๐ก is antisymmetric around t=0, thus only having non-zero ๐๐ โs
Time shifts in CTFS
6.003 Fall 2020
The time shifting in f(t) is carried to the underlying basis functions of it Fourier expansion.
Half-period shift inverts ๐๐ terms if k is odd. It has no effect if k is even.
๐ ๐ก =
๐ ๐ก = ฯ๐=0โ ๐๐
โฒ cos(๐๐0๐ก) + ฯ๐=1โ ๐๐
โฒ sin(๐๐0๐ก)
Half period shift
6.003 Fall 2020
The time shifting in f(t) is carried to the underlying basis functions of it Fourier expansion.
Half-period shift inverts ๐๐ terms if and only if k is odd.
๐ ๐ก =
๐ ๐ก = ฯ๐=0โ ๐๐
โฒ cos(๐๐0๐ก) + ฯ๐=1โ ๐๐
โฒ sin(๐๐0๐ก)
Quarter period shift?
Shifting by T /4 becomes more complicated.
โ ๐ก =
โ ๐ก = ฯ๐=0โ ๐๐
โฒโฒcos(๐๐0๐ก) + ฯ๐=1โ ๐๐
โฒโฒsin(๐๐0๐ก)
Quarter period shift?
6.003 Fall 2020
โ ๐ก =
โ ๐ก = ฯ๐=0โ ๐๐
โฒโฒcos(๐๐0๐ก) + ฯ๐=1โ ๐๐
โฒโฒsin(๐๐0๐ก)
Summary of shift result
6.003 Fall 2020
Other shifts will be even more complicated!
4โ ๐ก
Properties of CTFS: Time Shiftโข Consider ๐ฆ ๐ก = ๐ฅ ๐ก โ ๐ก0 , where ๐ฅ is periodic in ๐. What are the
CTFS coefficients ๐[๐], in terms of ๐[๐]?
๐๐๐ก ๐ข = ๐ก โ ๐ก0, then ๐ก = ๐ข + ๐ก0,d๐ก = ๐๐ข
Each coefficient ๐ ๐ in the series for ๐ฆ(๐ก) is a constant ๐โ๐๐๐0๐ times the corresponding coefficient ๐[๐] in the series for ๐ฅ(๐ก).
Properties of CTFS: Time Derivative
โข Consider ๐ฆ ๐ก =๐
๐๐ก๐ฅ ๐ก , where ๐ฅ ๐ก and ๐ฆ ๐ก are periodic in ๐. What
are the CTFS coefficients ๐[๐], in terms of ๐[๐]?
6.003 Fall 2020
Start with the synthesis equation:
Then, from the definition of y(ยท), we have:
Properties of CTFS: brief summary
โข Certain operations in the time domain have predictable effects in the frequency domain.
These can help us find the Fourier coefficients of a new signal without explicitly integrating!
If ๐ ๐ก is real valued periodic signal, ๐น ๐ = ๐นโ[โ๐]
The real part of F[k] comes from the symmetric part of the signal, the imaginary part of F k comes from the antisymmetric part of the signal
If ๐ฆ ๐ก = ๐ฅ โ๐ก , ๐[๐] = ๐[โ๐]
Check yourself!
Let ๐[๐] represent the Fourier series coefficients of the following signal:
6.003 Fall 2020
Check yourself!What is the relationship between the two following signals?
6.003 Fall 2020
๐[๐] = แ
0, ๐ ๐๐ ๐๐ฃ๐๐1
๐๐๐, ๐ ๐๐ ๐๐๐
Check yourself!The triangle waveform is the integral of the square wave.
๐ฅ ๐ก =๐
๐๐ก๐ฆ ๐ก
๐ ๐ = ๐2๐๐
๐๐ ๐ , ๐ = 1
๐[๐] = แ
0, ๐ ๐๐ ๐๐ฃ๐๐1
๐๐๐, ๐ ๐๐ ๐๐๐
๐ ๐ =1
๐2๐๐๐ ๐
๐ฆ ๐ก is symmetric around t=0, thus its Fourier Series coefficients are purely real
= แ0, ๐ ๐๐ ๐๐ฃ๐๐
โ1
2๐2๐2, ๐ ๐๐ ๐๐๐
Check yourself!
Let ๐[๐] represent the Fourier series coefficients of the following signal:
6.003 Fall 2020
๐[๐] = แ0, ๐ ๐๐ ๐๐ฃ๐๐
โ1
2๐2๐2, ๐ ๐๐ ๐๐๐
Summary
โข Various properties of the CTFS, making it easier to find the Fourier coefficients of a new signal
โข The Complex Exponential Form is much easier in math, will focus on this in the later part of the course
6.003 Fall 2020
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