To compare the performance of alternative filters, consider a square-wave signal with a frequency of f=0.33 cycles/min and an amplitude 0.5 corrupted by
(i) High-frequency sinusoidal noise (amplitude=0.25, $f_{N} =9$ cycles/min)
(ii) Random (Gaussian) noise with zero mean and a variance of 0.01
Evaluate both analog and digital single exponential filters, as well as a moving-average filter, and assess the effect of sampling interval Δt.

Question

To compare the performance of alternative filters, consider a square-wave signal with a frequency of f=0.33 cycles/min and an amplitude 0.5 corrupted by
(i) High-frequency sinusoidal noise (amplitude=0.25, [latex]f_{N} =9[/latex] cycles/min)
(ii) Random (Gaussian) noise with zero mean and a variance of 0.01
Evaluate both analog and digital single exponential filters, as well as a moving-average filter, and assess the effect of sampling interval Δt.

Accepted Answer

(i) Sinusoidal Noise
Representative results for high-frequency sinusoidal noise are shown in Fig.17.6. The square-wave with additive noise, the signal to be filtered, is shown in Fig. 17.6a, and the performance of two analog exponential filters with different filter time constants is shown in Fig. 17.6b. Choosing a relatively large filter time constant [latex] (τ_{F} =0.4 min) [/latex] results in a filtered signal that contains less noise but is more sluggish, compared to the response for [latex]τ_{F} =0.1 min. [/latex]
The effect of sampling period Δt on digital filter performance is illustrated in Fig. 17.6c. A larger sampling interval (Δt=0.1 min) results in serious aliasing, because [latex]f_{s} =1/Δt=10[/latex] cycles/min, which is less than [latex]2f_{N} =18[/latex] cycles/min for the added sinusoidal noise. Reducing Δt by a factor of two results in much better performance.
For each filter, a value of [latex]τ_{F} =0.1 min[/latex] was chosen, because this value was satisfactory for the analog filter of Fig. 17.6b. The smaller value of α(0.33 for Δt=0.05 min vs. 0.5 for Δt=0.1 min) provides more filtering.
The performance of two moving-average filters [latex](N^{∗} [/latex] =3 and 7) with Δt=0.05 min is shown in Fig. 17.6d.
Choosing [latex] N^{∗} =7 [/latex] results in better filtering because this moving-average filter averages the sinusoidal noise over several cycles, while [latex]N^{∗} =3[/latex] gives a faster response but larger fluctuations.

(ii) Random Noise
The simulations illustrating the effects of this noise level are shown in Fig. 17.7. Figure 17.7a shows the unfiltered signal after Gaussian noise with zero mean and a variance of 0.01 was added to the square-wave signal. The analog exponential filters in Fig. 17.7b provide effective filtering and again show the tradeoff between degree of filtering and sluggish response that is inherent in the choice of [latex]τ_{F} [/latex].
The digital filters in Fig. 17.7c and d are less effective, even though different values of Δt and [latex]N^{∗} [/latex] were considered. Some aliasing occurs owing to the high-frequency components of the random noise, which prevents the digital filter from performing as well as the analog filter.
In conclusion, both analog and digital filters can smooth noisy signals, providing that the filter design parameters (including sampling period) are carefully selected.

Question 17.1: To compare the performance of alternative filters, consider ...

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