It is frequently necessary to display on an oscilloscope screen waveforms having very s...

It is frequently necessary to display on an oscilloscope screen waveforms having very short time structures—for example, on the scale of thousandths of nanosecond. Since the use time of the fastest oscilloscope is longer than this, such display cannot be achieved directly. If however, the waveform is periodic, the desired result can be obtained indirectly by using an instrument called a sampling oscilloscope.

The idea, as shown in Figure P7.38(a), is to sample the fast waveform x(t) once each period, but at successively later points in successive periods. The increment ? should he an appropriately chosen sampling interval in relation to the bandwidth of x(t). If the resulting impulse train is then passed through an appropriate interpolating

lowpass filter, the output y(t) will be proportional to the original fast waveform slowed down or stretched out in time [i.e., y(t) is proportional to x(at), where a < 1].

For x(t) = A + B cos[(2π/T)t + θ], find a range of values of ? such that y(t) in Figure P7.38(b) is proportional to x(at) with a < 1. Also, determine the value of a in terms of T and ?.