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Constant, Sinusoid, Square, Triangle, and sawtooth waveforms, in depth and summarized.

Introduction

Once one has obtained a solid understanding of the fundamentals of Fourier series analysis and the General Derivation of the Fourier Coefficients , it is useful to have an understanding of the common signals used in Fourier Series Signal Approximation.

Deriving the fourier coefficients

Consider a square wave f(x) of length 1. Over the range [0,1), this can be written as

x ( t ) = 1 t 1 2 ; - 1 t > 1 2 .

Fourier series approximation of a square wave

Fourier series approximation to sq t . The number of terms in the Fourier sum is indicated in eachplot, and the square wave is shown as a dashed line over two periods.

Real even signals

Given that the square wave is a real and even signal,
  • f ( t ) = f ( - t ) EVEN
  • f ( t ) = f * ( t ) REAL
  • therefore,
  • c n = c - n EVEN
  • c n = c n * REAL

Consider this mathematical question intuitively: Can a discontinuous function, like the square wave, be expressed as asum, even an infinite one, of continuous signals? One should at least be suspicious, and in fact, it can't be thusexpressed.

The extraneous peaks in the square wave's Fourier series never disappear; they are termed Gibb's phenomenon after the American physicist Josiah Willard Gibbs. They occur whenever the signal isdiscontinuous, and will always be present whenever the signal has jumps.

Deriving the fourier coefficients for other signals

The Square wave is the standard example, but other important signals are also useful to analyze, and these are included here.

Constant waveform

This signal is relatively self-explanatory: the time-varying portion of the Fourier Coefficient is taken out, and we are left simply with a constant function over all time.

x ( t ) = 1

Sinusoid waveform

With this signal, only a specific frequency of time-varying Coefficient is chosen (given that the Fourier Series equation includes a sine wave, this is intuitive), and all others are filtered out, and this single time-varying coefficient will exactly match the desired signal.

x ( t ) = s i n ( π t )

Triangle waveform

x ( t ) = t t 1 / 4 2 - 4t 1 / 4 t 3 / 4 -7 / 4 + 4t 3 / 4 t 1
This is a more complex form of signal approximation to the square wave. Because of the Symmetry Properties of the Fourier Series, the triangle wave is a real and odd signal, as opposed to the real and even square wave signal. This means that
  • f ( t ) = - f ( - t ) ODD
  • f ( t ) = f * ( t ) REAL
  • therefore,
  • c n = - c - n
  • c n = - c n * IMAGINARY

Fourier series approximation of a triangle wave

Sawtooth waveform

x ( t ) = t - F l o o r ( t )
Because of the Symmetry Properties of the Fourier Series, the sawtooth wave can be defined as a real and odd signal, as opposed to the real and even square wave signal. This has important implications for the Fourier Coefficients.

Fourier series approximation of a sawtooth wave

Fourier series approximation vi

fourierSeriesDemo
Interact (when online) with a Mathematica CDF demonstrating the common Fourier Series. To download, right click and save file as .cdf.

link: http://yoder-3.institute.rose-hulman.edu/visible3/chapters/03spect/demosLV/fseries/index.htm

Summary

To summarize, a great deal of variety exists among the common Fourier Transforms. A summary table is provided here with the essential information.

Common continuous-time fourier series
Description Time Domain Signal for t [ 0 , 1 ) Frequency Domain Signal
Constant Waveform x ( t ) = 1 c k = 1 k = 0 0 k 0
Sinusoid Waveform x ( t ) = s i n ( π t ) c k = 1 / 2 k = ± 1 0 k ± 1
Square Waveform x ( t ) = 1 t 1 / 2 - 1 t > 1 / 2 c k = 4 / π k k odd 0 k even
Triangle Waveform x ( t ) = t t 1 / 2 1 - t t > 1 / 2 c k = - 8 Sin(kπ)/2) / ( π k ) 2 k odd 0 k even
Sawtooth Waveform x ( t ) = t / 2 c k = 0.5 k = 0 -1 / π k k 0

Questions & Answers

what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
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salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
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Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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