a periodic function f(t) is defined by
f(t) = t+pi -pi<t<0
f(t) = pi 0<t<pi
f(t) =f(t+2pi)
i) sketch the graph of the periodic function over the interval (-3pi, 3pi)
II) find the Fourier Series of f(t)
To sketch the graph of the periodic function over the interval (-3pi, 3pi), follow these steps:
1. Identify the important points:
- The function is linear in the interval (-pi, 0), with a positive slope.
- The function is constant at pi in the interval (0, pi).
2. Plot the important points:
- At t = -pi, the function value is 0.
- At t = 0, the function value is pi.
- At t = pi, the function value is also pi.
3. Determine the pattern:
- Since the function is periodic with a period of 2pi, the graph repeats itself every 2pi units.
- Therefore, you can copy the pattern from one interval, (-pi, 0), and repeat it for other intervals as well.
4. Sketch the graph:
- Start from t = -3pi and continue up to t = 3pi, copying the pattern every 2pi units.
- Connect the points smoothly to create a continuous graph.
The Fourier Series of f(t) can be found using the following steps:
1. Express f(t) as a periodic function:
- As given in the question, f(t) is already defined as a periodic function that repeats every 2pi units.
2. Determine the coefficients of the Fourier Series:
- The Fourier Series of a periodic function can be expressed as the sum of sine and cosine functions.
- To find the coefficients, evaluate the integral of the function multiplied by each corresponding sine or cosine term over one period.
- In this case, one period is 2pi.
3. Calculate the coefficient values:
- For n = 0, the constant term can be found by evaluating the integral of f(t) over one period and dividing by the length of the period (2pi).
- For n ≠ 0, the coefficients can be found by evaluating the integral of f(t) multiplied by cos(nωt) or sin(nωt) over one period and dividing by the length of the period.
4. Write the Fourier Series:
- The Fourier Series of f(t) can be expressed as the sum of the constant term (a0/2) and the series of cosine and sine terms, each multiplied by their respective coefficients.
Note: The steps mentioned above provide a generalized approach to finding the Fourier Series. The actual calculations may require more specific techniques depending on the function f(t).
I) To sketch the graph of the periodic function f(t) over the interval (-3π, 3π), we first need to plot the function for one period (0 to π) and then repeat it for subsequent periods.
1) For the interval -π < t < 0:
f(t) = t + π
Starting from -π, the graph of f(t) will increase linearly until it reaches 0 at t = 0. Then, it remains constant at the value π for the interval 0 < t < π.
2) Now, let's plot the function for the interval 0 < t < π:
f(t) = π
The value of f(t) is a constant π throughout this interval.
3) Next, we need to repeat this pattern for subsequent periods. Since f(t) is periodic with a period of 2π, we can draw the graph accordingly.
-- 0 -- π -- 2π -- 3π -- 4π -- 5π --
|->|
One period
By repeating this pattern and extending it further, we can sketch the graph of f(t) over the interval (-3π, 3π).
II) To find the Fourier series of f(t), we need to express it as a sum of sine and cosine terms.
In general, the Fourier series of a periodic function f(t) with a period of 2π is given by:
f(t) = (a0/2) + ∑[an * cos(nωt) + bn * sin(nωt)],
where n represents the harmonic index, ω is the angular frequency (ω = 2π), a0/2 is the constant term, and an and bn are the coefficients for the cosine and sine terms, respectively.
1) Constant term (a0/2):
The constant term for f(t) can be calculated by finding the average of the function over one period.
a0/2 = (1/π) * ∫[f(t)]dt over 0 to π
Since f(t) is a discontinuous function, we need to calculate the average separately for both intervals (-π to 0) and (0 to π).
Average over (-π, 0):
a0/2 = (1/π) * ∫[(t + π)]dt over -π to 0
= (1/π) * [(t^2/2 + πt)] evaluated at -π and 0
= (1/π) * [(0^2/2 + π(0))] - [(π^2/2 + π(-π))]
= (1/π) * (0 - [π^2/2 - π^2])
= (1/π) * (π^2/2) = π/2
Average over (0, π):
a0/2 = (1/π) * ∫[π]dt over 0 to π
= (1/π) * [πt] evaluated at 0 and π
= (1/π) * (π - 0) = 1
Thus, the constant term (a0/2) is π/2 for the interval (-π, 0) and 1 for the interval (0, π).
2) Coefficient for cosine term (an):
The coefficient for the cosine term can be calculated using the formula:
an = (2/π) * ∫[f(t) * cos(nωt)]dt over 0 to π
Calculating this integral for different values of n will give us the coefficients for the cosine terms.
3) Coefficient for sine term (bn):
The coefficient for the sine term can be calculated using the formula:
bn = (2/π) * ∫[f(t) * sin(nωt)]dt over 0 to π
Similarly, calculating this integral for different values of n will give us the coefficients for the sine terms.
By finding the respective coefficients for each harmonic index, the Fourier series of f(t) can be determined.