CH#3 Fourier Series and Transform King Saud University College of Applied studies and Community Service 1301CT By: Nour Alhariqi CH#3 Fourier Series and Transform 1st semester 1434-1435 nalhareqi_2013

Outline Introduction Fourier Series Fourier Series Harmonics Fourier Series Coefficients Fourier Series for Some Periodic Signals Example Fourier Series of Even Functions Fourier Series of Odd Functions Fourier Series- complex form nalhareqi_2013

Fourier Series Fourier Transform Introduction The Fourier analysis is the mathematical tool that shows us how to deconstruct the waveform into its sinusoidal components. This tool help us to changes a time-domain signal to a frequency-domain signal and vice versa. Time domain: periodic signal Frequency domain: discrete Fourier Series Time domain: nonperiodic signal Frequency domain: continuous Fourier Transform nalhareqi_2013

Fourier Series Fourier proved that a composite periodic signal with period T (frequency f ) can be decomposed into the sum of sinusoidal functions. A function is periodic, with fundamental period T, if the following is true for all t: f(t+T)=f(t) nalhareqi_2013

Fourier Series A periodic signal can be represented by a Fourier series which is an infinite sum of sinusoidal functions (cosine and sine), each with a frequency that is an integer multiple of f = 1/T  nalhareqi_2013

Fourier Series Harmonics Fourier Series = a sum of harmonically related sinusoids fundamental frequency the 2nd harmonic frequency the kth harmonic frequency the kth harmonic fundamental nalhareqi_2013

Fourier Series Harmonics ω ω ω ω ω ω nalhareqi_2013

Fourier Series Coefficients Are called the Fourier series coefficients, it determine the relative weights for each of the sinusoids and they can be obtained from DC component or average value nalhareqi_2013

Fourier Series for Some Periodic Signals nalhareqi_2013

Example The Fourier series representation of the square wave Single term representation of the periodic square wave nalhareqi_2013

Example The two term representation of the Fourier series of the periodic square wave The three term representation of the Fourier series of the periodic square wave nalhareqi_2013

Example Fourier representation to contain up to the eleventh harmonic nalhareqi_2013

Example Since the continuous time periodic signal is the weighted sum of sinusoidal signals, we can obtain the frequency spectrum of the periodic square-wave as shown below nalhareqi_2013

Example From the above figure we see the effect of compression in time domain, results in expansion in frequency domain. The converse is true, i.e., expansion in time domain results in compression in frequency domain. nalhareqi_2013

Even Functions The value of the function would be the same when we walk equal distances along the X-axis in opposite directions. t Mathematically speaking - nalhareqi_2013

Odd Functions The value of the function would change its sign but with the same magnitude when we walk equal distances along the X-axis in opposite directions. t Mathematically speaking - nalhareqi_2013

Fourier Series of Even Functions Even functions can solely be represented by cosine waves because, cosine waves are even functions. A sum of even functions is another even function. 10 5 q nalhareqi_2013

Fourier Series of Odd Functions Odd functions can solely be represented by sine waves because, sine waves are odd functions. A sum of odd functions is another odd function. 10 5 q nalhareqi_2013

Fourier Series of Even/Odd Functions The Fourier series of an even function is expressed in terms of a cosine series. The Fourier series of an odd function is expressed in terms of a sine series. nalhareqi_2013

Fourier Series- complex form The Fourier series can be expressed using complex exponential function The coefficient cn is given as nalhareqi_2013

Fourier Transform nalhareqi_2013

Outline Fourier transform Inverse Fourier transform Basic Fourier transform pairs Properties of the Fourier transform Fourier transform of periodic signal nalhareqi_2013

Fourier Transform Fourier Series showed us how to rewrite any periodic function into a sum of sinusoids. The Fourier Transform is the extension of this idea for non-periodic functions. the Fourier Transform of a function x(t) is defined by: The result is a function of ω (frequency).  nalhareqi_2013

Inverse Fourier Transform We can obtain the original function x(t) from the function X(ω ) via the inverse Fourier transform. As a result, x(t) and X(ω ) form a Fourier Pair: nalhareqi_2013

Example Let The called the unit impulse signal : The Fourier transform of the impulse signal can be calculated as follows So , x(t) t w X(w) nalhareqi_2013

Basic Fourier Transform pairs Often you have tables for common Fourier transforms nalhareqi_2013

Example Consider the non-periodic rectangular pulse at zero with duration τ seconds Its Fourier Transform is: nalhareqi_2013

properties of the Fourier Transform Linearity: Left or Right Shift in Time: Time Scaling: nalhareqi_2013

properties of the Fourier Transform Time Reversal: Multiplication by a Complex Exponential ( Frequency Shifting) : Multiplication by a Sinusoid (Modulation): nalhareqi_2013

Example: Linearity Let x(t) be : The Fourier Transform of x(t) will be : nalhareqi_2013

Example: Time Shift Let x(t) be : The Fourier Transform of x(t) will be : nalhareqi_2013

Example: Time Scaling time compression  frequency expansion time expansion  frequency compression nalhareqi_2013

Example: Multiplication by a Sinusoid Let x(t) be : The Fourier Transform of x(t) will be : nalhareqi_2013

Fourier Transform for periodic signal We learned that the periodic signal can be represented by the Fourier series as: We can obtain a Fourier transform of a periodic signal directly from its Fourier series the coefficient cn is given as nalhareqi_2013

Fourier Transform for periodic signal The resulting transform consists of a train of impulses in the frequency domain occurring at the harmonically related frequencies, which the area of the impulse at the nth harmonic frequency nω0 is 2π times nth the Fourier series coefficient cn So, the Fourier Transform is the general transform, it can handle periodic and non-periodic signals. For a periodic signal it can be thought of as a transformation of the Fourier Series nalhareqi_2013

Example Let The Fourier series representation of The Fourier series coefficients The Fourier transform of So, nalhareqi_2013

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Examples nalhareqi_2013

Example 1 Let which has the fundamental frequency Rewrite x(t) as a complex form and find the Fourier series coefficients ? Its known from Euler’s relation that: nalhareqi_2013

Solution Rewriting x(t) as a complex form, x(t) will be : nalhareqi_2013

Solution Thus, the Fourier series coefficients are: nalhareqi_2013

Example 2 Consider a periodic signal x(t) with fundamental frequency 2π, that has the following Fourier series coefficients Rewrite x(t) as a trigonometric form? From the given coefficients, the x(t) in complex form nalhareqi_2013

Solution rewriting x(t) and collecting each of the harmonic components which have the same fundamental frequency, we obtain Using Euler’s relation, x(t) can be written as: nalhareqi_2013

Example 3 A periodic signal x(t) with a fundamental period T = 8 has the following nonzero Fourier series coefficients Express x(t) in the trigonometric form? The fundamental frequency is nalhareqi_2013

Example 3 Let , find its Fourier transform ? The Fourier series representation of is The Fourier series coefficients The Fourier transform of is nalhareqi_2013