[link] shows how this
sequence of signals portrays the signal more accuratelyas more terms are added.
Fourier series spectrum of a half-wave rectified sine wave
The Fourier series spectrum of a half-wave rectified sinusoidis shown in the upper portion. The index indicates the
multiple of the fundamental frequency at which the signal hasenergy. The cumulative effect of adding terms to the Fourier
series for the half-wave rectified sine wave is shown in thebottom portion. The dashed line is the actual signal, with
the solid line showing the finite series approximation to theindicated number of terms,
.
We need to assess quantitatively the accuracy of theFourier series approximation so that we can judge how rapidly
the series approaches the signal. When we use a
-term series, the error—the difference between
the signal and the
-term series—corresponds to the unused terms from
the series.
To find the rms error, we must square this expression and
integrate it over a period. Again, the integral of mostcross-terms is zero, leaving
[link] shows how the error in the
Fourier series for the half-wave rectified sinusoid decreases asmore terms are incorporated. In particular, the use of four
terms, as shown in the bottom plot of
[link] , has a rms error (relative
to the rms value of the signal) of about 3%. The Fourier seriesin this case converges quickly to the signal.
Approximation error for a half-wave rectified sinusoid
The rms error calculated according to
[link] is shown as a function of the number of terms in theseries for the half-wave rectified sinusoid.
The error has been normalized by the rms value of thesignal.
We can look at
[link] to
see the power spectrum and the rms approximation error for thesquare wave.
Power spectrum and approximation error for a square wave
The upper plot shows the power spectrum of the square wave,
and the lower plot the rms error of the finite-lengthFourier series approximation to the square wave. The
asterisk denotes the rms error when the number of terms
in the Fourier series equals 99. Because the Fourier coefficients decay more slowly here than for
the half-wave rectified sinusoid, the rms error is notdecreasing quickly. Said another way, the square-wave's
spectrum contains more power at higher frequencies than does thehalf-wave-rectified sinusoid. This difference between the two
Fourier series results because the half-wave rectifiedsinusoid's Fourier coefficients are proportional to
while those of the square wave are proportional to
. If fact, after 99 terms of the square wave's
approximation, the error is bigger than 10 terms of theapproximation for the half-wave rectified sinusoid.
Mathematicians have shown that no signal has an rmsapproximation error that decays more slowly than it does for the
square wave.
Questions & Answers
What are the factors that affect demand for a commodity
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
When MP₁ becomes negative, TP start to decline.
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 •
Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Economic growth as an increase in the production and consumption of goods and services within an economy.but
Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has
The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50.
A,Calculate quantities of x and y which maximize utility.
B,Calculate value of Lagrange multiplier.
C,Calculate quantities of X and Y consumed with a given price.
D,alculate optimum level of output .
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
Abdureman
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