Fourier Transform and Spectra Chapter 2 Fourier Transform and Spectra Topics: Rectangular and Triangular Pulses Spectrum of Rectangular, Triangular Pulses Convolution Spectrum by Convolution Huseyin Bilgekul Eeng360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University

Rectangular Pulses

Triangular Pulses

Spectrum of a Rectangular Pulse Rectangular pulse is a time window. FT is a Sa function, infinite frequency content. Shrinking time axis causes stretching of frequency axis. Signals cannot be both time-limited and bandwidth-limited. Note the inverse relationship between the pulse width T and the zero crossing 1/T

Spectrum of Sa Function To find the spectrum of a Sa function we can use duality theorem. Duality: W(t)  w(-f) Because Π is an even and real function

Spectrum of a Time Shifted Rectangular Pulse The spectra shown in previous slides are real because the time domain pulse (rectangular pulse) is real and even. If the pulse is offset in time domain to destroy the even symmetry, the spectra will be complex. Let us now apply the Time delay theorem of Table 2.1 to the Rectangular pulse. T 1 Time Delay Theorem: w(t-Td)  W(f) e-jωTd We get:

Spectrum of a Triangular Pulse The spectrum of a triangular pulse can be obtained by direct evaluation of the FT integral. An easier approach is to evaluate the FT using the second derivative of the triangular pulse. First derivative is composed of two rectangular pulses as shown. The second derivative consists of the three impulses. We can find the FT of the second derivative easily and then calculate the FT of the triangular pulse.

Spectrum of a Triangular Pulse

Spectrum of Rectangular and Sa Pulses

Table 2.2 Some FT pairs

Key FT Properties Time Scaling; Contracting the time axis leads to an expansion of the frequency axis. Duality Symmetry between time and frequency domains. “Reverse the pictures”. Eliminates half the transform pairs. Frequency Shifting (Modulation); (multiplying a time signal by an exponential) leads to a frequency shift. Multiplication in Time Becomes complicated convolution in frequency. Mod/Demod often involves multiplication. Time windowing becomes frequency convolution with Sa. Convolution in Time Becomes multiplication in frequency. Defines output of LTI filters: easier to analyze with FTs. x(t) x(t)*h(t) h(t) X(f) X(f)H(f) H(f)

Convolution The convolution of a waveform w1(t) with a waveform w2(t) to produce a third waveform w3(t) which is where w1(t)∗ w2(t) is a shorthand notation for this integration operation and ∗ is read “convolved with”. If discontinuous wave shapes are to be convolved, it is usually easier to evaluate the equivalent integral Evaluation of the convolution integral involves 3 steps. Time reversal of w2 to obtain w2(-λ), Time shifting of w2 by t seconds to obtain w2(-(λ-t)), and Multiplying this result by w1 to form the integrand w1(λ)w2(-(λ-t)).

Example for Convolution For 0< t < T For t > T

y(t)=x(t)*z(t)=  x(τ)z(t- τ )d τ Convolution y(t)=x(t)*z(t)=  x(τ)z(t- τ )d τ Flip one signal and drag it across the other Area under product at drag offset t is y(t). x(t) z(t) z(t-t) x(t) z(t) t t t -1 1 t -1 1 t t-1 t t+1 z(-2-t) z(-6-t) z(0-t) z(2-t) z(4-t) -6 -4 -2 -1 1 2 t 2 y(t) -6 -4 -2 -1 1 2 t