# 2.2 Dsp00108-averaging time series  (Page 8/14)

 Page 8 / 14

## A short recap before continuing

Hopefully, by this point, you understand how multiplying two time series produces a new time series composed of the sum of all the products of theindividual sinusoids in the two original time series.

When each pair of sinusoids is multiplied together, they produce a new time series consisting of two other sinusoids whose frequencies are the sum anddifference of the original pair of frequencies.

## The error in the computed average

When an average is computed for a fixed number of points on the new time series, the error in the average tends to be greater for cases where theoriginal frequency values were close together. This is because the period of one of the new sinusoids becomes longer as the original frequencies become closer.In general, the longer the period of the sinusoid, the more points are requiredto get a good estimate of its average value.

## Does this matter?

There are many operations in DSP where this matters a lot. As mentioned earlier, the computational requirements for DSP frequently boil down to nothingmore than multiplying a pair of time series and computing the average of the product. You will see many examples of this as you continue studying the modulesin this series of tutorials on DSP.

## Spectral analysis

I am going to illustrate my point by showing you one such example in this module. This example will use a Fourier transform in an attempt to performspectral analysis and to separate two closely-spaced frequency components in a time series. As you will see, errors in the computed average can interfere withthis process in a significant way.

(This example will illustrate and explain the results using graphs. Future modules will provide more technical details on the DSP operationsinvolved.)

## Several steps are involved

I will provide this illustration in several steps.

## Spectral data for same frequency but different lengths

First, I will show you spectral data for several time series, each consisting of a single sinusoid. The time series will have different lengths but theindividual sinusoids will have the same frequency. This will serve as baseline data for the experiments that follow.

## Sum of two sinusoids

Then I will show you spectral data for several time series, each composed of the sum of two sinusoids. These time series will have different lengths. Thesinusoids in each time series will have the same frequencies. I will show you two cases that fall under this description. The frequency difference for the twosinusoids in each time series will be small in one case, and greater in another case.

## Sinusoids with different frequency differences

Finally, I will show you spectral data for several time series, each composed of the sum of two sinusoids. These time series will be different lengths, andthe sinusoids in each time series will have different frequencies. In particular, the frequency difference between the two sinusoids in each timeseries will be equal to the theoretical frequency resolution for a time series of that particular length.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
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
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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