<< Chapter < Page
  Digital signal processing - dsp     Page 8 / 14
Chapter >> Page >
If you know the value of a single real sample and you know its position in the series relative to the origin, you can write equationsthat describe the real and imaginary parts of the transform of that single sample without any requirement to actually perform a Fourier transform.

Those equations are simple sine and cosine equations as a function of the units of the output domain. This is an important concept that contributesgreatly to the implementation of the FFT algorithm.

Transformation of a complex series

The FFT algorithm is an algorithm that transforms a series of complex values in one domain into a series of complex values in another domain. The images inthe figures discussed so far indicate a transformation of a complex function given by f(x) into another complex function given by F(k). There is nothing inthese images to indicate anything about time and frequency.

If the complex part of the input series f(x) is not zero, things get somewhat more complicated. For example, the real and imaginary parts of the transform ofan impulse having both real and imaginary parts are not necessarily cosine and sine curves. This is illustrated in Figure 8 .

Figure 8. Transform of a complex impulse with a shift equal to two sample intervals.
missing image

Figure 8 shows the results of transforming an impulse having both real andimaginary parts and a shift of two sample intervals.

Although both the real and imaginary parts of the transformed result have the shape of a sinusoid, neither is a cosine curve and neither is a sine curve. Bothof the curves are sinusoidal curves that have been shifted along the horizontal output axis moving their peaks and zero crossings away from the origin.

Linearity still applies

Because the Fourier transform is a linear transform, you can transform the real and imaginary parts of the input separately and add the two resultingtransforms. The sum of the two transforms represents the transform of the entire input series including both real and imaginary parts. The program that I willdiscuss later takes advantage of this fact. Once again, the main point is:

Even for a complex input series, if you know the values of the real and imaginary parts of a sample and you know the value of the shiftassociated with that sample, you can write equations that describe the real part and the imaginary part of the transform results.

Can produce the transform of a time series by the adding transforms of the individual samples

That brings us to the crux of the matter. Given an input series consisting of a set of sequential samples taken atuniform sampling intervals, we know how to write equations for the real and imaginary parts that would be produced by performing a Fourier transform oneach of those samples individually.

The input series is the sum of the individual samples

We know that we can consider the input series to consist of the sum of the individual samples, each having a specified value and a different shift. We knowthat the Fourier transform is a linear transform. Therefore, the Fourier transform of an input series is the sum of the transforms of the individualsamples.

Questions & Answers

What do we mean by Asian tigers
Aeesha Reply
Dm me I will tell u
Shailendra
Hi
Aeesha
hi
Pixel
What is Average revenue
KEMZO
How are u doing
KEMZO
it is so fantastic
metasebia
what's a demand
Edward Reply
it is the quantity of commodities that consumers are willing and able to purchase at particular prices and at a given time
Munanag
quantity of commodities dgat consumers are willing to pat at particular price
Omed
demand depends upon 2 things 1wish to buy 2 have purchasing power of that deserving commodity except any from both can't be said demand.
Bashir
Demand is a various quantity of a commodities that a consumer is willing and able to buy at a particular price within a given period of time. All other things been equal.
Vedzi
State the law of demand
Vedzi
The desire to get something is called demand.
Mahabuba
what is the use of something should pay for its opportunity foregone to indicate?
Random Reply
Why in monopoly does the firm maximize profits when its marginal revenue equals marginal cost
astrid Reply
different between economic n history
Falma Reply
If it is known that the base change of RM45 million, the statutory proposal ratio of 7 per cent, and the public cash holding ratio of 5 per cent, what is the proposed ratio of bank surplus to generate a total deposit of RM 300 million? 
Jeslyne Reply
In a single bank system, a bank can create a deposit when it receives a new deposit in cash. If a depositor puts a cash deposit of RM10,000 into the bank, assume the statutory reserve requirement is 7% and the bank adopts a surplus reserve of 8%. a. Calculate the amount of deposits made at the end o
Jeslyne
the part of marginal revenue product curve lies in the _ stage of production is called form demand curve for variable input.
Bashir Reply
The cost associated with the inputs owned by the farmer is termed as
Bashir
the cost associated with inputs owned by the farmer is termed as ____
Bashir
why do we study economic
Nwobodo Reply
we study economics to know how to manage our limited resources
Eben
တစ်ဦးကျဝင်​ငွေ
myo
we study economics the know how to use our resources and where to put it
Mamoud
what is end
Nwobodo
we study economics to make rational decision
Gloria
we study economics only to know how to effectively and efficiently allocate our limited resource in other to meet our unlimited wants
Kpegba
We study economics inorder for us to know the difference of the needs and wants and aslo how to use the limited resources that are available
Bongani
who is the father of economy
Yajanyi Reply
adam smith
ibrahim
Adam smith
Somnath
professor Lionel Robins
Abraham
adam smith
albert
mariginal utility is finalized by who?
Barsharani
marshall
Aadi
Adam Smith
Laila
Adam smith
Mamoud
Adam Smith
Bongani
Adam smith
Veronica
adam smith barter system
Omed
why we study economics
Kitojo Reply
what is equilibrium price?
BENJAMIN Reply
This is the price In which quantity demanded is equal to the quantity supplied.
Binta
what is the principle of demand
Udoh
is when the price of two item is equal
Mamoud
is the market price at which the demand curve and supply curve of particular commodity interest.
Kpegba
can we say that without macroeconomics,microeconomics can succeed? and why?
Kpegba
equilibrium price is when prices are equal
Ramon
equilibrium price is a point at which demand and supply curve meet
Vedzi
please can you give us the correct answer after the lesson to be compared to our answers
Gloria Reply
in what?
ibrahim
why economics is the real life subject
Kitojo
because it is subjected to human decisions
Kpegba
why might an increase in money national income not necssarily lead to an increase in the standards of living
Multin
pls,who is a legal tender.can you explain well
Mary Reply
We think, that the legal tender is a form of payment of a debt or anything related, but which is not necessarily money. that can be bank notes, or coins for instance. but the bottom line is the legal tender is required to be recognized by the law, but it varies according to the jurisdiction.
Pierre
Thanks
Mary
Is it something like cheque
Mary
legal tender is anything that can be accepted for payment within a country
Tyson
good
Philo
is Something legally accepted in a particular place
Mamoud
Who is a liquidator?
Isah Reply
pls,who is a legal tender
Mary
We think, that the legal tender is a form of payment of a debt or anything related, but which is not necessarily money. that can be bank notes, or coins for instance. but the bottom line is the legal tender is required to be recognized by the law, but it varies according to the jurisdiction.
Pierre
s and D are the only in the firm.the face following playoff D gain: large budget. small Bud 30billon. 0 zero 70billon. 50milln s gains large budg. small bud 20millon. 0zero 30. 40
Mujahid Reply
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 Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play




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