# 2.2 Dsp00108-averaging time series  (Page 7/14)

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## An expanded view

Figure 6 shows a greatly expanded view of the red average values in Figure 5 .

Figure 6. Expanded average value of a time series. The ideal value for this average is 0.5, and that is the value represented by the blue line. The plot in Figure 6 shows the same horizontal scale as Figure 5 . However, the entire vertical plotting area in Figure 6 represents the values from 0.48 to 0.52.

## Ideal value is never reached

As you can see, the ideal value is never reached in Figure 6 except at isolated points where the red curve crosses the horizontal line. Even if Iextended the horizontal axis to 1200 or more points, that would continue to be the case.

## A more serious case

Figure 7 computes and displays the average value of the bottom plot in Figure 2 (recall that this plot shows 1200 points on the horizontal axis, whereas Figure 5 shows only 400 points on the horizontal axis). Recall also that this time series was produced by multiplying two sinusoids having nearly thesame but not exactly the same frequency.

Figure 7. Computed average value of a time series. ## Red curve is the average

As before, the black curve in Figure 7 shows the time series, and the red curve shows the computed average value as a function of the number of pointsincluded in the average.

(In this case, I didn't even bother to show the short axis containing only 400 points. The horizontal axis in Figure 7 contains 1200 points, the same as in Figure 2 .)

## The ideal average value is zero

In this case, the ideal average value is zero, as indicated by the green horizontal axis. As you can see, even for a 1200-point averaging window, theaverage value deviates significantly from the ideal. We will see the detrimental impact of this problem later when I perform spectral analysis in an attempt toseparate two closely-spaced peaks in the frequency spectrum.

## Some additional examples of average values

Figure 8 computes and displays the average value of the third plot down from the top in Figure 3 . This plot was produced by multiplying the two sinusoids in the top two plots in Figure 3 .

Figure 8. Computed average value of a time series. As before, the black curve in Figure 8 represents the time series, and the red curve represents the computed average value of the time series as a functionof the number of points included in the average.

For this case also, the ideal average value is zero, as represented by the green horizontal axis. The positive and negative peaks in the red average valuecan be seen to approach the ideal value asymptotically within the 400 horizontal points plotted in Figure 8 .

Figure 9 computes and displays the average value of the bottom plot in Figure 3 . This time series was produced by multiplying the top plot in Figure 3 by the fourth plot in Figure 3 .

Figure 9. Computed average value of a time series. Once again, the black curve in Figure 9 represents the time series, and the red curve represents the computed average value of the time series as a functionof the number of points included in the average. In this case, the average converges on zero rather nicely within the 400 points included on the horizontalaxis.

explain and give four Example hyperbolic function
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
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combine like terms. x + x + 2 is same as 2x + 2
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x*x=2
felecia
2+2x=
felecia
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how do I set up the problem?
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hi mam
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use the y -intercept and slope to sketch the graph of the equation y=6x
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hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
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Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
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Need help solving this problem (2/7)^-2
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the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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