Data Acquisition

exact replicas of itself spaced about multiples of the sampling frequency. Figure 5.14(d)illustrates the frequency spectrum of a signal that is sampled at exactly twice the maximumfrequency of the original signal, showing that the replicas of the original signal just touch.Oversampling the original signal, as shown in Figure 5.14(e) separates the input signal bandsby a wider frequency. This is shown in Figure 5.14(f). Undersampling narrows the separationbetween the bands so that they fold over each other and result in aliasing, asdemonstrated in Figure 5.14(g) and Figure 5.14(h). Where this occurs, the resultant signalappears as an aliased signal between DC and the Nyquist frequency, and cannot bedistinguished from valid data.f(t)F( T )tT 02T 02Ta) Time-varying band limited signal b) Frequency Spectrumf(t)F( T )TT 2T 3T 8Tt2T 0T 02BTT 0 2 T 0Tc) Signal sampled at Nyquist rate d) Spectral replicas generated when sampling at Nyquist ratef(t)F( T )T 2T 3T 8TtT 2e) Over sampling a time varying signal f) Spectral replicas generated when over sampling2T 0T 04BTT 0 2T 0Tf(t)F( T )2T 4T 6T 8TtT 02T 02Tg) Under sampling a time varying signal h) Spectral replicas generated when under samplingFigure 5.14Demonstrating the effect of aliasing in the frequency domainConsider a band-limited signal, which contains three sinusoidal waveforms, a 25 Hz waveformrepresenting the wanted signal, a 50 Hz signal, which is unwanted mains hum, and anunwanted high frequency noise signal at 260 Hz. Figure 5.15(a) shows the frequencyspectrum of this band-limited signal.