A single cycle of a ramp function of voltage versus time

A single cycle of a ramp function of voltage versus time has the form v( t) = lOOt from -0.01 to 0.01 s. Using direct integration, evaluate the Fourier coefficients ao, at. a2, bt. and b2. Could you have deduced the values of ao, aI, and a2 without performing the integrations?

A single cycle of a ramp function of voltage versus time has the form v(t) = SOt from -0.02 to 0.02 s. Using direct integration, evaluate the Fourier coefficients ao, aI, a2, bI, and b2 Could you have deduced the values of ao, aI, and a2 without performing the integrations?

A single cycle of a ramp function has the form v(t) = 25t from t = 0 to t = 1 s. Evaluate the Fourier constants ao, a1> a2, bt. and hz by direct integration.

A single cycle of a ramp function has the form v(t) = lOt from t = 0 to t = 2 s. Evaluate the Fourier constants ao, aI, a2, bI, and hz by direct integration.

For the ramp function described in 5.1, evaluate the coefficients ao, a1. a2, b1 and b2 using a numerical method such as described in Section Al. Use 100 equally spaced time intervals.

For the ramp function described in Problem 5.5, evaluate the coefficients ao, at. a2, bt. and hz using a numerical method such as described in Section Al. Use 100 equally spaced time intervals.

A transient voltage has the form v(t) = 20t - 200t2 in the time interval 0-0.1 s. Specify a periodic form of this function so that only sine terms will be required for a Fourier-series approximation

A transient voltage has the form v(t) = 20t - 100t2 in the time interval 0-0.2 s. Specify a periodic form of this function so that only sine terms will be required for a Fourier-series approximation.

A transient voltage has a value of 3.5 V from t = 0 to t = 1.5 s and has a value of zero at other times. Specify a periodic form of this function so that it can be represented entirely with a Fourier cosine series.

A transient voltage has a value of 5 V from t = 0 to t = 2 s and has a value of zero at other times. Specify a periodic form of this function so that it can be represented entirely with a Fourier cosine series.

The function shown in Figure P5.7 can be represented by the Fourier cosine series 2 00 ( _1)n f(t) = 0.2 + ---sin 0.2mr cos 4mrt 11" n=l n Using a spreadsheet program, compute the sum of ao, at. a2, and a3 terms at subintervals of 0.01 s over the interval 0 to 0.5 s. Plot this sum and compare it to the original function.

Using a spreadsheet program as shown in Section A.2, generate 128 values of the function f(t) = 5 sin 101I"t + 3 cos 401l"t between t = 0 and t = 1 s. Perform an FFT on the results and find the magnitude of the coefficients (as produced by the FFT; they will be complex). Plot the results versus frequency in a bar chart such as Figure 5.12. Interpret the peaks observed. *

Using a spreadsheet program as shown in Section A.2, generate 128 values of the function f(t) = 3 cos 5001l"t + 5 cos 8oo1l"t between t = 0 and t = 0.1 s. Perform an FFT on the results and find the magnitude of the coefficients (as produced by the FFT, they will be complex). Plot the results versus frequency in a bar chart such as Figure 5.14. Interpret the peaks observed.

Consider the function f(t) = 7.5 sin2 101I"t. Using a spreadsheet program as shown in Section A.2, generate 128 values between t = 0 and t = 2 s. Perform a FFT on the results, evaluate the magnitude of the coefficients, and present the result in a bar graph such as Figure 5.14. What is the meaning of the coefficient at f = O?

Using a spreadsheet program, generate 256 values of the equation f(l) = 3 sin(211"16.7t) + 2 cos(211"33.4t) between t = 0 and 1 second. In the next column, generate 256 values of the Hann window function (Eq. 5.15) and in a fourth column, generate the product of the second and third columns. Plot the original function and the windowed function on the same graph. Perform an FFT on the original results and take the steps to create a bar graph like Figure 5.15. Perform an FFT on the windowed data and create another bar graph. Compare the two bar graphs and discuss the difference.

A function has the form f(t) = 5 sin 101Tt + 3 cos 401Tt and is sampled at a rate of 30 samples per second. What false alias frequency(ies) would you expect in the discrete data?

A function has the form f(t) = 6 sin 201Tt + 5 cos 301Tt and is sampled at a rate of 50 samples per second. What false alias frequency(ies) would you expect in the discrete data?

The function in Problem 5.16 is sampled at 5 samples per second. What false alias frequencies would you expect in the discrete data?

What would be the minimum sampling rate to avoid false aliasing for the function in Problem 5.16?

The function f(t) = 3 cos 5001Tt + 5 cos 8001Tt is sampled at 400 samples per second starting at t = 0.00025 s. What false alias frequencies would you expect in the output?

The function f(t) = 5 cos 1oo1Tt + 90 cos 5OO1Tt is sampled at 200 samples per second starting at t = 0.00025 s. What false alias frequencies would you expect in the output?

What would be the minimum sampling rate to avoid false frequencies in the sampled data for the function in Problem 5.20?

Using a spreadsheet program, plot the function of Problem 5.20 from t = 0.00025 to 0.1 s at 0.0025-s intervals. Connect the points with straight lines. Explain the shape of the resulting plot

A 3.5-kHz sine wave signal is sampled at 2 kHz. What would be the lowest expected alias frequency?

A 5-kHz sine wave signal is sampled at 3 kHz. What would be the lowest expected alias frequency?

A I-kHz sine wave is sampled at 1.5 kHz. What is the lowest expected alias frequency?

A 3-kHz sine wave is sampled at 4 kHz. What is the lowest expected alias frequency?

What is the dynamic range of a 16-bit bipolar NO converter?

What is the dynamic range of a 14-bit unipolar NO converter?

For an experiment, fc is 10 kHz and the NO converter is bipolar with 12 bits. If a secondorder Butterworth lowpass filter is used, what will fm be if the signal is attenuated to the converter threshold?

In a vibration experiment, an acceleration-measuring device can measure frequencies up to 3 kHz, but the maximum frequency of interest is 500 Hz. A data acquisition system is available with a 12-bit bipolar NO converter and a maximum sampling rate of 10,000 samples per second. Specify the corner frequency and order of a Butterworth antialiasing filter for this application (attenuating the signal to the converter threshold) and the actual minimum sampling rate.