# 2.2 Dsp00108-averaging time series  (Page 11/14)

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(I also applied an additional scale factor to the spectral results to compensate for the fact that fewer total samples were included in theaverage for the short samples. This caused the amplitude of the peak in the spectrum to be nominally the same in all five cases.)

## A horizontally-expanded plot

As you can see in Figure 10 , there isn't much in the spectra to the right of about 50 spectral points. That is as it should be since the single sinusoid ineach time series was at the low end of the spectrum.

Figure 11 shows the same data as Figure 10 with only the first 50 frequency points plotted on the horizontal axis. The remaining 350 frequency points weresimply ignored. This provides a much better view of the structure of the peaks in the different spectra.

Figure 11. Spectra of five different sinusoids of different lengths.

I will begin the discussion with the bottom plot in Figure 11 , which is the computed spectrum for the single sinusoid having a length of 400 samples.

## A spectral line

Ideally, since the time series was a single sinusoid, the spectrum should consist of a single non-zero value at the frequency of the sinusoid, (often referred to as a spectral line) and every other value in the spectrum should be zero.

However, because the computation of the spectrum involves the computation of average values resulting from the products of sinusoids, the ideal is not alwaysachieved. In order to achieve the ideal, it would be necessary to multiply and average over an infinite number of points. Anything short of that will result insome measurement error, as exhibited by the bottom plot in Figure 11 .

(The bottom plot in Figure 11 has a large peak in the center with every second point to the left and right of center having a zero value. Iwill explain this structure in more detail later.)

## Spectra of shorter sinusoids

Moving from the bottom to the top in Figure 11 , each individual plot shows the result of shorter and shorter averaging windows. As a result, themeasurement error increases and the peak broadens for each successive plot going from the bottom to the top in Figure 11 . The plot at the top, with an averaging window of only 80 samples, exhibits the most measurement error and the broadestpeak.

(It should be noted, however, that even the spectra for the shorter averaging windows have some zero-valued points. Once you understand thereason for the zero-valued points, you can correlate the positions of those points to the length of the averaging windows in Figure 11 .)

## Two spectral lines

Now I'm going to show you the detrimental impact of such spectral measurement errors. In particular, the failure of the average to converge on zero for shortaveraging windows limits the spectral resolution of the Fourier transform.

## Five time series with two sinusoids each

I will create five new time series, each consisting of the sum of two sinusoids with fairly closely-spaced frequencies. One sinusoid has 32 samplesper cycle as in Figures 10 and 11. The other sinusoid has 26 samples per cycle.

What is demand
the amount of a good that buyers are willing and able to purchase
Asit
what is population
The people living within a political or geographical boundary.
Ziyodilla
what happens to price and quantity when demand curves shift to the right
price level goes up. quantity demand increases
Asit
example- inferior goods
Asit
demand law
Athony
Its states that higher the price the of the commodity, and lower the quantity demanded
Kosiso
I am confused but quantity demand will increase.
Asit
No. That's the law of supply
Kosiso
the price goes down and quantity demanded increases
esther
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price level will increase
Asit
quantity demand will decrease
Asit
what is inflation
inflation is a general and ongoing rise in the level of prices in an entire economy.
cynthia
is the pasistance increase in the price of a country economy
Liyu
kk
Duppy
yes
how does inflation affects the economy of a country? what is deflation?
Augustine
deflation can simply be define as the persistence decrease in price of a countrys economy
Liyu
the revenge of malthus relates "revenge" with "commodity prices". collect data for 3 commodoties and check their price evolution
what is elasticity
Elasticity is an economics concept that measures responsiveness of one variable to changes in another variable.
cynthia
right
Augustine
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cynthia
Computer software represents
पर्यावरण राज्यों में से किस राज्य में शिष्य शिक्षक अनुपात 30 से अधिक वाले विद्यालयों का प्रतिशत न्यूनतम होता है
Hey what are you trying to mean?
Kenyana
what is Asset
MUBARAK
like a banana
Ahmed
demand is the process whereby consumers are willing and able to purchase a particular product at various price over a given period of time
The law of dinimish
What is the law of dinimish
Frank
What is the law of dinimish
Frank
What is the law of dinimish
Frank
opportunity cost is to forgo something for another.
yes
King
what is financial market
what is demand
Demand is an economic principle referring to a consumer's desire to purchase goods and services and willingness to pay a price for a specific good or service.
Ali
explain any three exceptions to the law of demand
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