6.3 Properties of the ctfs

Introduction

In this module we will discuss the basic properties of the Continuous-Time Fourier Series. We will begin by refreshing your memory of our basic Fourier series equations:

Linearity

$ℱ(·)$ is a linear transformation .

If $ℱ(f(t))=c_n$ and $ℱ(g(t))=d_n$ . Then $$\forall \alpha , \alpha \in \mathbb{C}\colon ℱ(\alpha f(t))=\alpha c_n$$ and $$ℱ(f(t)+g(t))=c_n+d_n$$

Shifting

Shifting in time equals a phase shift of Fourier coefficients

$ℱ(f(t-t_0))=e^{-(i{\omega }_{0}n{t}_0)}{c}_n$ if $c_n=\left|c_n\right|e^{i\angle (c_n)}$ , then $$\left|e^{-(i{\omega }_{0}n{t}_0)}{c}_n\right|=\left|e^{-(i{\omega }_{0}n{t}_0)}\right|\left|c_n\right|=\left|c_n\right|$$ $$\angle (e^{-(i{\omega }_0{t}_{0}n)})=\angle (c_n)-{\omega }_0{t}_{0}n$$

Parseval's relation

The rate of decay of the Fourier series determines if $f(t)$ has finite energy .

Parsevals theorem demonstration

Symmetry properties

Even signals

Even signals

• $f\left(t\right)=f(-t)$
• $\parallel c_n\parallel =\parallel c_{-n}\parallel$

• $$c_n=\frac{1}{T}{\int }_0^{T}f\left(t\right)exp(-ı{\omega }_{0}nt)dt$$
• $$=\frac{1}{T}{\int }_0^{\frac{T}{2}}f\left(t\right)exp(-ı{\omega }_{0}nt)dt+\frac{1}{T}{\int }_{\frac{T}{2}}^{T}f\left(t\right)exp(-ı{\omega }_{0}nt)dt$$
• $$=\frac{1}{T}{\int }_0^{\frac{T}{2}}f(-t)exp(-ı{\omega }_{0}nt)dt+\frac{1}{T}{\int }_{\frac{T}{2}}^{T}f(-t)exp(-ı{\omega }_{0}nt)dt$$
• $$=\frac{1}{T}{\int }_0^{T}f\left(t\right)\left[exp,(ı{\omega }_{0}nt),,d,t,+,exp,(-ı{\omega }_{0}nt)\right]dt$$
• $$=\frac{1}{T}{\int }_0^{T}f\left(t\right)2cos\left({\omega }_{0}nt\right)dt$$

Odd signals

Odd signals

• $f\left(t\right)=\mathrm{-f}\left(\mathrm{-t}\right)$
• $c_n=c_{-n}$ *

• $$c_n=\frac{1}{T}{\int }_0^{T}f\left(t\right)exp(-ı{\omega }_{0}nt)dt$$
• $$=\frac{1}{T}{\int }_0^{\frac{T}{2}}f\left(t\right)exp(-ı{\omega }_{0}nt)dt+\frac{1}{T}{\int }_{\frac{T}{2}}^{T}f\left(t\right)exp(-ı{\omega }_{0}nt)dt$$
• $$=\frac{1}{T}{\int }_0^{\frac{T}{2}}f\left(t\right)exp(-ı{\omega }_{0}nt)dt-\frac{1}{T}{\int }_{\frac{T}{2}}^{T}f(-t)exp(ı{\omega }_{0}nt)dt$$
• $$=-\frac{1}{T}{\int }_0^{T}f\left(t\right)\left[exp,(ı{\omega }_{0}nt),,d,t,-,exp,(-ı{\omega }_{0}nt)\right]dt$$
• $$=-\frac{1}{T}{\int }_0^{T}f\left(t\right)2ısin\left({\omega }_{0}nt\right)dt$$

Real signals

Real signals

• $f\left(t\right)=f$ * $\left(t\right)$
• $c_n=c_{-n}$ *

• $$c_n=\frac{1}{T}{\int }_0^{T}f\left(t\right)exp(-ı{\omega }_{0}nt)dt$$
• $$=\frac{1}{T}{\int }_0^{\frac{T}{2}}f\left(t\right)exp(-ı{\omega }_{0}nt)dt+\frac{1}{T}{\int }_{\frac{T}{2}}^{T}f\left(t\right)exp(-ı{\omega }_{0}nt)dt$$
• $$=\frac{1}{T}{\int }_0^{\frac{T}{2}}f(-t)exp(-ı{\omega }_{0}nt)dt+\frac{1}{T}{\int }_{\frac{T}{2}}^{T}f(-t)exp(-ı{\omega }_{0}nt)dt$$
• $$=\frac{1}{T}{\int }_0^{T}f\left(t\right)\left[exp,(ı{\omega }_{0}nt),,d,t,+,exp,(-ı{\omega }_{0}nt)\right]dt$$
• $$=\frac{1}{T}{\int }_0^{T}f\left(t\right)2cos\left({\omega }_{0}nt\right)dt$$

Differentiation in fourier domain

Since

Fourier differentiation demonstration

Integration in the fourier domain

If

Integration accentuates low frequencies and attenuates high frequencies. Integrators bring out the general trends in signals and suppress short term variation (which is noise in many cases). Integrators are much nicer than differentiators.

Fourier integration demonstration

Signal multiplication and convolution

Given a signal $f(t)$ with Fourier coefficients $c_n$ and a signal $g(t)$ with Fourier coefficients $d_n$ , we can define a new signal, $y(t)$ , where $y(t)=f(t)g(t)$ . We find that the Fourier Series representation of $y(t)$ , $e_n$ , is such that $e_n=\sum_{k=()}$∞ ∞ c k d n - k . This is to say that signal multiplication in the time domainis equivalent to signal convolution in the frequency domain, and vice-versa: signal multiplication in the frequency domain is equivalent to signal convolution in the time domain.The proof of this is as follows

Conclusion

Like other Fourier transforms, the CTFS has many useful properties, including linearity, equal energy in the time and frequency domains, and analogs for shifting, differentation, and integration.