EE3TP4_13_FourierSeriesDetails_v2_Lecture 17

# EE3TP4_13_FourierSeriesDetails_v2_Lecture 17 - Fourier...

This preview shows pages 1–8. Sign up to view the full content.

Fourier Analysis

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
3.2 & 3.3 Fourier Series Expansion In the last set of notes we looked at building signals using: x t = k =− N N c k e jk ω 0 t N = finite integer In general we need an infinite number of terms: x t = A 0 k = 1 N A k cos 0 t θ k x t = k =−∞ c k e jk ω 0 t Fourier Series (Complex Exponential Form) x t = A 0 k = 1 A k cos 0 t θ k Fourier Series (Trigonometric Form) These are 3 different forms for the same expression. x t = a 0 k = 1 [ a k cos 0 t  b k sin 0 t ] Fourier Series (Trigonometric Form)
“ . .. the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour. “ Jean Baptiste Joseph Fourier About the Fourier Series Expansion Laplace Lagrange

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Q: Does this now let us form any periodic signal? A: No! Although Fourier thought so! Dirichlet showed that there are some that can’t be written in terms of a FS! Sufficient conditions for the FS expansion to exist . .. The function must 1. have a finite number of extrema in any given interval, 2. have a finite number of discontinuities in any given interval, 3. be absolutely integrable over a period, 4. be bounded. But … those will never show up in practice! So we can write any practical periodic signal as a FS with infinite # of terms!
We can now break virtually any periodic signal into a sum of simple things… and we already understand how these simple things travel through an LTI system! So, instead of: h t x t y t = x t ∗ h t We break x ( t ) into its FS components and find how each component goes through. (See Chapter 5)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
To do this kind of convolution-evading analysis we need to be able to solve the following: Given time-domain signal model x ( t ) Find the FS coefficients { c k } “Time-domain” model “Frequency-domain model” Converting “time-domain” signal model into a “frequency-domain” signal model
Fourier Series x t = k =−∞ c k e jk ω 0 t c k = 1 T t 0 t 0 T x t e jk ω 0 t dt T = 2π/ω 0 ω 0 = fundamental frequency (rad/sec) This is true for a very wide class of periodic signals!

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 23

EE3TP4_13_FourierSeriesDetails_v2_Lecture 17 - Fourier...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online