SIGNAL RECONSTRUCTION

8.2 SIGNAL RECONSTRUCTION

The process of reconstructing a continuous-time signal x (t) from its samples is also known as interpolation. In Section 8.1 , we saw that

a signal x (t) bandlimited to B Hz can be reconstructed (interpolated) exactly from its samples if the sampling frequency f s exceeds 2B Hz or the sampling interval T is less than 1/2B. This reconstruction is accomplished by passing the sampled signal through an ideal

lowpass filter of gain T and having a bandwidth of any value between B and f s − B Hz. From a practical viewpoint, a good choice is the middle value f s /2= 1/2T Hz or π/T rad/s. This value allows for small deviations in the ideal filter characteristics on either side of the cutoff frequency. With this choice of cutoff frequency and gain T, the ideal lowpass filter required for signal reconstruction (or

interpolation) is

The interpolation process here is expressed in the frequency domain as a filtering operation. Now we shall examine this process from the time-domain viewpoint.

TIME-DOMAIN VIEW: A SIMPLE INTERPOLATION Consider the interpolation system shown in Fig. 8.5a . We start with a very simple interpolating filter, whose impulse response is rect

(t/T), depicted in Fig. 8.5b . This is a gate pulse centered at the origin, having unit height, and width T (the sampling interval). We shall find the output of this filter when the input is the sampled signal x (t) consisting of an impulse train with the nth impulse at t= nT with strength x (nT). Each sample in x (t), being an impulse, produces at the output a gate pulse of height equal to the strength of the sample. For instance, the nth sample is an impulse of strength x (nT) located at t= nT and can be expressed as x (nT) δ(t − nT). When this impulse passes through the filter, it produces at the output a gate pulse of height x (nT), centered at t= nT (shaded in Fig. 8.5c) . Each sample in x (t) will generate a corresponding gate pulse, resulting in the filter output that is a staircase approximation of x (t), shown dotted in Fig. 8.5c . This filter thus gives a crude form of interpolation.

Figure 8.5: Simple interpolation by means of a zero-order hold (ZOH) circuit. (a) ZOH interpolator. (b) Impulse response of a ZOH circuit. (c) Signal reconstruction by ZOH, as viewed in the time domain. (d) Frequency response of a ZOH.

The frequency response of this filter H ( ω) is the Fourier transform of the impulse response rect (t/T).

and

The amplitude response |H ( ω)| for this filter, illustrated in Fig. 8.5d , explains the reason for the crudeness of this interpolation. This filter, also known as the zero-order hold (ZOH) filter, is a poor form of the ideal lowpass filter (shaded in Fig. 8.5d) required for exact interpolation. [ † ]

We can improve on the ZOH filter by using the first-order hold filter, which results in a linear interpolation instead of the staircase interpolation. The linear interpolator, whose impulse response is a triangle pulse Δ(t/2T), results in an interpolation in which successive sample tops are connected by straight-line segments (see Prob. 8.2-3).

TIME-DOMAIN VIEW: AN IDEAL INTERPOLATION The ideal interpolation filter frequency response obtained in Eq. (8.8) is illustrated in Fig. 8.6a . The impulse response of this filter, the

inverse Fourier transform of H ( ω) is

Figure 8.6: Ideal interpolation for Nyquist sampling rate.

For the Nyquist sampling rate, T= 1/2B, and

This h (t) is depicted in Fig. 8.6b . Observe the interesting fact that h (t)= 0 at all Nyquist sampling instants (t= ±n/2B) except at t= 0. When the sampled signal x (t) is applied at the input of this filter, the output is x (t). Each sample in x (t), being an impulse, generates

a sinc pulse of height equal to the strength of the sample, as illustrated in Fig. 8.6c . The process is identical to that depicted in Fig. 8.5c , except that h (t) is a sinc pulse instead of a gate pulse. Addition of the sinc pulses generated by all the samples results in x (t). The nth sample of the input x (t) is the impulse x (nT) δ(t−nT); the filter output of this impulse is x (nT)h (t−nT). Hence, the filter output to x (t), which is x (t), can now be expressed as a sum

For the case of Nyquist sampling rate, T= 1/2B, and Eq. (8.11a) simplifies to

Equation (8.11b) is the interpolation formula, which yields values of x (t) between samples as a weighted sum of all the sample values.

EXAMPLE 8.3

Find a signal x (t) that is bandlimited to B Hz, and whose samples are where the sampling interval T is the Nyquist interval for x (t), that is, T= 1/2B.

Because, we are given the Nyquist sample values, we use the interpolation formula (8.11b) to construct x (t) from its samples. Since all but one of the Nyquist samples are zero, only one term (corresponding to n= 0) in the summation on the right-hand side of Eq. (8.11b) survives. Thus

This signal is illustrated in Fig. 8.6b . Observe that this is the only signal that has a bandwidth B Hz and the sample values x (0)= 1 and x (nT)= 0 (n ≠ 0). No other signal satisfies these conditions.