Skip to content Skip to navigation

OpenStax_CNX

You are here: Home » Content » Zero-Order Hold

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

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 δt δt , we might convolve with a rectangular pulse of width TT, known as a zero-order hold (ZOH) function and denoted by hzt hz t . This yields xzt xz t :

xzt=n =xnhz tnT xz t n xn hz t n T
(1)
Unwanted spectral copies in Xz iΩ 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 δt 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 iω= hz te(iΩt)d t =0T1Te(iΩt)dt =sinΩT2ΩT2e(iΩT2) 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 iΩ Hr Ω must invert the ZOH response Hz iΩ Hz Ω over the passband πT πT T T . The left side of Figure 3 shows the ideal |Hr iΩ| 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 iΩ 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)

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks