Digital filtering

Description

In digital filtering, a Fourier transform is performed on a noisy spectrum to convert it to the frequency domain. The resulting power spectrum is then multiplied by a filter function to remove all undesired frequency components. (The filter function is equal to 0 at all frequencies to be removed, and equal to 1 at all other frequencies.) An inverse Fourier transform then converts the data back to the original domain.

In this simulation, a time-based signal is simulated by adding together two sine waves with user-defined frequencies, amplitudes, and phase difference, along with Gaussian white noise. The power spectrum is calculated and displayed. The cutoff frequencies of a band-pass or notch filter are entered, and the original data is displayed with all frequencies outside of the filter removed.

(Reference: Skoog, Holler, and Crouch, Principles of Instrumental Analysis, 6th Ed, Thomson Brooks/Cole 2007)

Exercises and questions

• 1. Make a sketch of the unfiltered signal that results with the following conditions:

     • • Sine Wave #1: Freq = 30, Amp = 2

       • Sine Wave #2: Freq = 5, Amp = 1, Phase = 90

       • Noise: 2

  2. Now make sketches of the filtered signal and explain what is happening when using bandpass cutoffs of:

     • • 0 and 100

       • 0 and 10

       • 4 and 6

       • 29 and 31

  3. What generalizations can you make regarding the effect of bandpass width on the resulting filtered signal?

  4. What effect does increasing or decreasing the noise level have on the filtered signal?

  5. What is the difference between a notch and band-pass filter?