MECH 322 Instrumentation Performed: 03/15/06 Differentiation and Spectral Analysis of Discretely Sampled Signals Group 0 Pablo Araya Lab Instructors: Mithun Gudipati, Venkata Venigalla

ABSTRACT The objective of this lab is to measure sine and sawtooth wave voltage signals using a data acquisition system and demonstrate errors that can occur when determining their time derivative and spectral content. Time derivatives were determined using first order finite differencing with different time steps. The random noise decreased as the time step increased but the accuracy of following step changes in slope decreased. The peak in the measured spectral content was in agreement with signal frequency when the sampling frequency was at least twice the signal frequency.

Table. 1 Sine and Sawtooth Wave Parameters Peak to peak voltage and maximum frequency are determined from scope and frequency counter. The maximum slopes of the ideal wave forms are (dV/dt) max,sine =  V PP f M, and (dV/dt) max,sawtooth = 2V PP f M Input resolution error W v depends on A/D converter voltage range (±5V) and number of bits (N = 14) and W v =

Fig.1 Sine Wave and Derivative Based on Different Time Steps dV/dt 1 (  t= sec) is nosier than dV/dt 10 (  t= sec) The maximum slope from the finite difference method is larger than the ideal value. This may be because the actual wave was not a pure sinusoidal.

Fig. 2 Sawtooth Wave and Derivative Based on Different Time Steps dV/dt 1 is again nosier than dV/dt 10 dV/dt 1 responds to the step change in slope more accurately than dV/dt 10 The maximum slope from the finite difference method is larger than the ideal value.

Fig. 3 Measured Spectral Content of 100 Hz Sine Wave for Different Sampling Frequencies The measured peak frequency f P equals the maximum signal frequency f M = 100 Hz when the sampling frequency f S is greater than 2f M f s = 70 and 150 Hz do not give accurate indications of the peak frequency.

Table 2 Peak Frequency versus Sampling Frequency For f S > 2f M = 200 Hz the measured peak is close to f M. For f S < 2f M the measured peak is close to the magnitude of f M –f S. The results are in agreement with sampling theory.

Fig. 4 VI Front Panel

Figure 5 VI Block Diagram

Table 3 Signal and Indicated frequency data (extra credit) This table shows the dimensional and dimensionless signal frequency f m (measured by scope) and frequency indicated by spectral analysis, f a. For a sampling frequency of f S = 48,000 Hz, the folding frequency is f N = 24,000 Hz.

Figure 6 Dimensionless indicated frequency versus signal frequency (extra credit) The characteristics of this plot are similar to those of the textbook folding plot For each indicated frequency f a, there are many possible signal frequencies, f m.