Filter Effect in Digital Data Processing

Abstract

This chapter deals with filter effects in digital data processing. For this purpose the transfer function is introduced. Simple filters like high-pass, low-pass, band-pass, and notch are discussed. The effects of data shifting, data compression as well as differentiation and integration of discrete data are shown.

Playground

5.1

Totally Different

Given is the function $f(t)=\cos (\pi t/2)$ which is sampled at times $t_k=k \Delta t$, $k=0,1,\ldots ,5$ with $\Delta t = 1/3$.

Calculate the first central difference and compare it with the “exact” result for $f'(t)$. Plot your results! What is the percentage error?

5.2

Simpson’s-1/3 versus Trapezoid

Given is the function $f(t)=\cos \pi t$ which is sampled at times $t_k=k \Delta t$, $k=0,1,\ldots ,4$ with $\Delta t = 1/3$.

Calculate the integral using the Simpson’s $1/3$-rule and the Trapezoidal Rule and compare your results with the exact value.

5.3

Totally Noisy

Given is a cosine input series that’s practically smothered by noise (Fig. 5.16).

$$\begin{aligned} f_i=\cos \frac{\pi j}{4} + 5(\text {RND} - 0.5), \qquad j = 0, 1, \dots , N. \end{aligned}$$

(5.47)

In our example, the noise has a $2.5$-times higher amplitude than the cosine signal. (The signal-to-noise ratio (power!) therefore is $0.5:5/12=1.2$, see playground 4.6.)

In the time spectrum (Fig. 5.16) we can’t even guess the existence of the cosine component.

Fig. 5.16

Cosine signal in totally noisy background according to (5.47)

Fig. 5.17

Discrete line on slowly falling background

• a. What Fourier transform do you expect for series (5.47)?

• b. What can you do to make the cosine component visible in the time spectrum, too?

5.4

Inclined Slope

Given is a discrete line as input, that’s sitting on a slowly falling ground (Fig. 5.17).

• a. What’s the most elegant way of getting rid of the background?

• b. How do you get rid of the “undershoot”?