<< Chapter < Page
  Digital signal processing - dsp     Page 3 / 14
Chapter >> Page >

Suppose, for example, that we have two time series, each of which is composed of two sinusoidal components as follows:

f(x) = cos(ax) + cos (bx) g(x) = cos(cx) + cos(dx)

The product of the two time series is given by:

h(x) = f(x)*g(x) = (cos(ax) + cos (bx)) * (cos(cx) + cos(dx))

where the asterisk (*) means multiplication.

Multiplying this out produces the following:

h(x) = cos(ax)*cos(cx) + cos(ax)*cos(dx)+ cos(bx)*cos(cx) + cos(bx)*cos(dx)

A sum of products of sinusoids

Thus, the time series produced by multiplying any two time series consists of the sum of a (potentially large) number of terms, each of which is the product of two sinusoids.

The product of two sinusoids

We probably need to learn a little about the product of two sinusoids. I will discuss this topic with a little more mathematical rigor in a future module. Inthis module, however, I will simply illustrate the topic using graphs.

Important: The product of two sinusoids is always a new time series, which is the sum of two new sinusoids.

The frequencies of the new sinusoids

The frequencies of the new sinusoids are different from the frequencies of the original sinusoids. Furthermore, the frequency of one of the new sinusoidsmay be zero.

What is a sinusoid with zero frequency?

As a practical matter, a sinusoid with zero frequency is simply a constant value. It plots as a horizontal straight lineversus time.

Think of it this way. As the frequency of the sinusoid approaches zero, the period, (which is the reciprocal of frequency), approaches infinity. Thus, the width of the first lobe of the sinusoid widens, causing every value in thatlobe to be the same as the first value.

This will become a very important concept as we pursue DSP operations.

Sum and difference frequencies

More specifically, when you multiply two sinusoids, the frequency of one of the sinusoids in the new time series is the sum of the frequencies of the two sinusoids that were multiplied together. The frequency of the othersinusoid in the new time series is the difference between the frequencies of the two sinusoids that were multiplied together.

An important special case

For the special case where the two original sinusoids have the same frequency, the difference frequency is zero and one of the sinusoids in the newtime series has a frequency of zero. It is this special case that makes digital filtering and digital spectrum analysis possible.

Many sinusoidal products

When we multiply two time series and compute the average of the resulting time series, we are in effect computing the average of the products of all theindividual sinusoidal components contained in the two time series. That is, the new time series contains the products of (potentially many) individual sinusoids contained in the two original time series. In the end, it all comesdown to computing the average value of products of sinusoids.

Product of sinusoids with same frequency

The product of any pair of sinusoids that have the same frequency will produce a time series containing the sum of two sinusoids. One of the sinusoidswill have a frequency of zero (hence it will have a constant value). The other sinusoid will have a frequency that is double the frequency of theoriginal sinusoids.

Questions & Answers

the art of managing the production, distribution and consumption.
Satangthem Reply
what is economics
Khawar Reply
what is Open Market Operation
Adu Reply
dominating middlemen men activities circumstances
Christy Reply
what Equilibrium price
Adji Reply
what is gap
mirwais
who is good with the indifference curve
Dexter
What is diseconomic
Alixe Reply
what are the types of goods
WARIDI
how can price determination be the central problem of micro economics
simon Reply
marginal cost formula
Nandu Reply
you should differentiate the total cost function in order to get marginal cost function then you can get marginal cost from it
boniphace
What about total cost
Foday
ok
Foday
how can price determination be the central problem if micro economics
simon
formula of cross elasticity of demand
Theresia Reply
what is ceteris paribus
Priyanka Reply
what is ceteris parabus
Priyanka
Ceteris paribus - Literally, "other things being equal"; usually used in economics to indicate that all variables except the ones specified are assumed not to change.
Abdullah
What is broker
scor
land is natural resources that is made by nature
scor
What is broker
scor
what is land
kafui
What is broker
scor
land is natural resources that is made by nature
scor
whats poppina nigga turn it up for a minute get it
amarsyaheed Reply
what is this?
Philo
am from nigeria@ pilo
Frank
am from nigeria@ pilo
Frank
so
owusu
what is production possibility frontier
owusu
it's a summary of opportunity cost depicted on a curve.
okhiria
please help me solve this question with the aid of appropriate diagrams explain how each of the following changes will affect the market price and quantity of bread 1. A
Manuela Reply
please l need past question about economics
Prosper Reply
ok let me know some of the questions please.
Effah
ok am not wit some if den nw buh by tommorow I shall get Dem
adepojurafiu
Hi guys can I get Adam Smith's WEALTH OF NATIONS fo sale?
Ukpen
hello I'm Babaisa alhaji Mustapha. I'm studying Economics in the university of Maiduguri
Babaisa
okay
Humaira
my name is faisal Yahaya. i studied economics at Kaduna state university before proceeding to West African union university benin republic for masters
Faisal
Hi guys..I am from Bangladesh..
Mannan
Wat d meaning of management
igwe Reply
disaster management cycle
Gogul Reply
cooperate social responsibility
igwe
Fedric Wilson Taylor also define management as the act of knowing what to do and seeing that it is done in the best and cheapest way
OLANIYI
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing - dsp' conversation and receive update notifications?

Ask