Create the graph in excel For the following wave functions, assume amplitude A = 20 mm and force frequency f = 10

Create the graph in excel

 

 

For the following wave functions, assume amplitude A = 20 mm and force frequency ω_f = 10 Hz (= 62.8 rad/sec). Set up your time signal equations so that each sine wave component is in a separate column cell. This will let you add up different combinations of the cells by row as needed for plotting the various Fourier series. Rows should vary by time increment (i.e., if the time increment is 0.05 sec: for row 1, t = 0; for row 2, t = 0.05 sec; for row 3, t = 0.10 sec, etc.).

 

"SIGNALS" you may use: 

Square Wave, SQ(t) = A sin ω_ft + A/3 sin 3ω_ft + A/5 sin 5ω_ft + A/7 sin 7ω_ft + A/9 sin 9ω_ft + A/11 sin 11ω_ft + ... 

 

Triangle Wave, TR(t) = A*4/(PI)2 * [sin 1ω_ft - 1/9 sin 3ω_ft + 1/25 sin 5ω_ft - 1/49 sin 7ω_ft + 1/81 sin9ω_ft - 1/121 sin 11ω_ft...]

 

On sections 1 through 4, plot either square or triangle waves, not both. In either case, the equation consists of sine terms at the odd multiples of force frequency, multiplied by a constant amplitude value. The level of difficulty for setting up the triangle wave is a bit higher so 3 bonus points will be assigned for choosing it. 

NOTE: 

 

The Fourier Series representation of a time signal must generally neglect some of the higher frequency portions of the signal.  This approximation may or may not be sufficient to represent the signal accurately. 

 

Procedure: 

1. The accuracy of a time domain signal depends on whether the significant frequency components (the larger amplitude components) are included. 

 

Generate a plot with two versions of your wave (square or triangle) on EXCEL. For the first curve, use the sum of the four lowest frequency sine wave components; for the second curve, use the sum of all six of the listed sine wave components. Use a time interval of 0.004 seconds for both plots (0.000, 0.004, 0.008, etc.), with TIME being the leftmost column of the spreadsheet. Set up the EXCEL file so that each sine wave component is located in its own column, as shown below. This allows easy summation of values from different combinations of sine wave components, with the summation columns to the right of the component columns. Make sure your curves are identified. Each waveform should display a minimum of one full signal cycle. 

 

Time

Sine ωf term

Sine  3ωf term

Sine  5ωf term

Sine  7ωf term

Sine  9ωf term

Sine  11 ωf term

Sum   of four sine terms

Sum   of six sine terms

 

1. Time, frequency, and amplitude can be considered to be three orthogonal axes.  The plots in section 1 show a time-domain view of your wave, with axes of time and amplitude.  In order to view the signal in the frequency domain, a Fourier Transformation must be made. Visually, you can extract the same information from the motion equations (if the equations exist). BONUS points available if you figure out how to get EXCEL to do the Fourier transformation so you have similar results (10 points maximum). 

 

            Generate one plot -the amplitude and frequency of your six sine wave component equation on a column graph in the frequency domain. 

 

           Amplitude 

 

 

 

Frequency, rad/sec 

 

1. The ability to "recover" the original signal from a digitally sampled signal depends on the rate of sampling (or conversely, the sampling time interval).  The minimum acceptable sampling frequency (rate) is twice the value of the highest frequency component of the signal.   This is known as the Nyquist Criterion.  The distortion seen on your first plot will be due to aliasing. 

 

            Generate a two-period plot with three versions of your six sine wave component equation. (1) Use a sampling time interval of 0.05 seconds. (2) Use a sampling time interval that corresponds to the Nyquist Criterion. (3) Use a sampling time interval that is one-half or less than the Nyquist Criterion interval used on your previous plot. (Your spreadsheet from section 1 still applies, with the exception of time interval changes for each of these three plots). Each waveform should display a minimum of one full signal cycle. 

 

            The Nyquist Criterion indicates that data must be sampled fast enough to be at twice the frequency (or half the period) of the highest useful frequency component in the data or faster. 

 

5.         Filtering removes portions of the measured signal. This gives a mechanism for setting a maximum frequency (as is needed to prevent aliasing) and for removing low-frequency noise. Unfortunately, it also provides a mechanism for inadvertently removing useful and needed portions of the signal when applied improperly. 

 

            Generate two plots (column and scatter). Apply a bandpass filter with a highpass limit of 65 rad/sec and a lowpass limit of 650 rad/sec to the frequency plot from section 2.  Plot your remaining frequencies, then generate and plot a time waveform based on the remaining frequencies.  Compare to your six-term time waveform plot from section 1. 

 

            The amplitude versus frequency plot will correspond to the EXCEL column graph set up in section 2.  The amplitude versus time plot will be set up like the plots in sections 1 and 3, and can use the section 1 spreadsheet columns (with the omission of any sine terms that have been "filtered" out).