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Zero-Order Hold

Module by: Phil Schniter. E-mail the author

Summary: Introduction to the use of zero order hold.

Note: In practice, it is not possible to generate Dirac deltas for the creation of xs ⁢t xs t . So, instead of convolving with δ⁢t δt , we might convolve with a rectangular pulse of width TT, known as a zero-order hold (ZOH) function and denoted by hz⁢t hz t . This yields xz⁢t xz t :

xz⁢t=∑n =−∞∞x⁢n⁢hz ⁢t−n⁢T xz t n xn hz t n T
(1)
Unwanted spectral copies in Xz ⁢i⁢Ω Xz Ω can be removed by a final stage of lowpass filtering. The two-step procedure is illustrated in Figure 1. As with the Dirac delta, we assume a ZOH function with unit area, requiring that the analog reconstruction lowpass filter has DC gain T T.

Figure 1: ZOH reconstruction of Nyquist-bandlimited signal.
Figure 1 (zoh.png)

Unlike convolution with δ⁢t δt , convolution with the ZOH function hz ⁢t hz t introduces passband "droop" and out-of-band attenuation. This results from the fact that the frequency response magnitude of the ZOH function is not constant in Ω:

H z ⁢i⁢ω=∫−∞∞ hz ⁢t⁢e−(i⁢Ω⁢t)d t =∫0T1T⁢e−(i⁢Ω⁢t)dt =sinΩ⁢T2Ω⁢T2⁢e−(i⁢Ω⁢T2) H z ω t hz t Ω t t 0 T 1 T Ω t Ω T 2 Ω T 2 Ω T 2
(2)
(See Figure 2 for an illustration.) Thus, for perfect reconstruction, the analog reconstruction filter Hr ⁢i⁢Ω Hr Ω must invert the ZOH response Hz ⁢i⁢Ω Hz Ω over the passband −πT πT T T . The left side of Figure 3 shows the ideal |Hr ⁢i⁢Ω| Hr Ω .

Figure 2: ZOH function.
Figure 2 (boxcar.png)

It is difficult to build analog reconstruction filters with sharp cutoff. Instead, one hopes that the desired signal is bandlimited to less than the Nyquist frequency (as in Figure 1), so that there is an absence of unwanted spectral energy in a region around ±⁢πT ± T . In this case, the analog reconstruction filter Hr ⁢i⁢Ω Hr Ω may be designed with a wider transition band, such as on the right side of Figure 3.

Figure 3: Ideal(left) and practical(right) reconstruction filters for ZOH.
Figure 3 (recon_filter.png)

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