{"id":1927,"date":"2026-08-16T20:07:16","date_gmt":"2026-08-16T20:07:16","guid":{"rendered":"https:\/\/smartdata.ece.ufl.edu\/?p=1927"},"modified":"2026-08-17T00:45:49","modified_gmt":"2026-08-17T00:45:49","slug":"inverse-z-transform","status":"publish","type":"post","link":"https:\/\/smartdata.ece.ufl.edu\/index.php\/2026\/08\/16\/inverse-z-transform\/","title":{"rendered":"Inverse Z-Transform"},"content":{"rendered":"\n

Motivation<\/h2>\n\n\n\n

One of the challenges with Z-transform is that that inverse is not unique. A single z-transform representation can correspond have multiple time-domain signals. However, if you know the z-transform’s region of convergence, you can unique identify the inverse.<\/p>\n\n\n\n

Region of Convergence<\/h3>\n\n\n\n

Since the Z-transform is a sum, then \"X(z)=\infty\" if any value in the sum is \"\infty\". This is not invertible (i.e., we cannot get back to \"x[n]\"). Hence, we can only invert for values of \"z\" in which \"X(z) \neq \infty\". This set of \"z\" values is known as the region of convergence<\/strong>. The region of convergence exists in the complex plane.<\/p>\n\n\n\n

In the complex plane, the region of convergence begins at a pole. If the pole is causal,<\/strong> the region of convergence exists for all values with a radius greater than \"\sigma\" (the magnitude of the pole) from the origin.<\/p>\n\n\n\n

If the pole is anti-causal,<\/strong> the region of convergence exists for all values with a radius of less than \"\sigma\" (the magnitude of the pole) from the origin. If the system has multiple poles,<\/strong> the total region of convergence is the intersection of the region of convergence for each pole.<\/p>\n\n\n\n

An Example<\/h3>\n\n\n\n

Consider

  <\/span>   <\/span>\"\[x[n] = (1/2)^n u[n] \; .\]\"<\/p> The Z-transform is defined by

  <\/span>   <\/span>\"\[X(z) = \frac{1}{1-(1/2)z^{-1}} \; .\]\"<\/p> The time-domain signal \"x[n]\" is causal, so the ROC is \"|z| > 1/2\".<\/p>\n\n\n\n

Similarly, consider

  <\/span>   <\/span>\"\[x[n] = -(1/2)^n u[-n-1] \; .\]\"<\/p> The Z-transform is defined by

  <\/span>   <\/span>\"\[X(z) = \frac{1}{1-(1/2)z^{-1}} \; .\]\"<\/p> The time-domain signal \"x[n]\" is anti-causal, so the ROC is \"|z| < 1/2\".<\/p>\n\n\n\n

Show Proof<\/strong><\/summary>\n

Consider

  <\/span>   <\/span>\"\[x[n] = -(1/2)^n u[-n-1] \; .\]\"<\/p>
The time-reversal of \"x[n]\" is

  <\/span>   <\/span>\"\[x[-n] = -(1/2)^{-n} u[n-1] = -(2)^{n} u[n-1] \; .\]\"<\/p>
The Z-transform of \"x[-n]\" is \"X(z^{-1})\" due to the time-reversal property. To compute the Z-transform of a time-domain signal, we can make one additional modification:

  <\/span>   <\/span>\"\[x[-(n+1)] = -(2)^{n+1} u[n] = -2 (2)^{n} u[n] \; .\]\"<\/p>
By applying the time-shifting property, the Z-transform of \"x[-(n+1)]\" is \"X(z^{-1}) z^{+1}\".<\/p>\n\n\n\n

Hence, by using the Z-transform table, we can establish that

  <\/span>   <\/span>\"\[X(z^{-1}) z^{+1} = -2 \frac{1}{1 - 2 z^{-1}} \; .\]\"<\/p>
This can be simplified to

  <\/span>   <\/span>\"\[<span class="ql-right-eqno"> </span><span class="ql-left-eqno"> </span><img src="https://smartdata.ece.ufl.edu/wp-content/ql-cache/quicklatex.com-3edf431baa77f6e2a5c3ebbcf9333b04_l3.png" height="45" width="567" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align*}X(z^{-1}) &= -2 \frac{z^{-1}}{1- 2 z^{-1}} \X(z) &= -2 \frac{z}{1 - 2 z} \&= -2 \frac{1}{z^{-1} - 2} \&= \frac{1}{1 - (1/2) z^{-1}}\end{align*}" title="Rendered by QuickLaTeX.com"/>\]\"<\/p><\/p>\n<\/details>\n\n\n\n

Partial Fraction Decomposition<\/h2>\n\n\n\n

One of the most common ways to perform an inverse Z-transform representation is to decompose it into easy-to-invert components. Partial fraction decomposition for \"N > M\" (i.e., more poles than zeros) can be expressed as <\/p>\n\n\n\n

  <\/span>   <\/span>\"\begin{align*} H(z) &= \frac{b_0}{a_0} \frac{\prod_{m=1}^{M} \left( 1 - z_{m} z^{-1} \right) }{\prod_{m=1}^{N} \left( 1 - p_{m} z^{-1} \right) } \\ &= \frac{b_0}{a_0} \sum_{m=1}^N \frac{A_m}{1 - p_{m} z^{-1} } \; , \end{align*}\"<\/p><\/p>\n\n\n\n

where \"p_{m}\" are scalar constants representing our poles and \"A_m\" are scalar constants for each term. After performing the partial fraction decomposition, we can complete the inverse Z-transform for each component in \"H(z)\".<\/p>\n","protected":false},"excerpt":{"rendered":"

Motivation One of the challenges with Z-transform is that that inverse is not unique. A single z-transform representation can correspond have multiple […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"post-templates\/post_noheader_nofooter2.php","format":"standard","meta":{"footnotes":""},"categories":[76,108,1],"tags":[],"class_list":["post-1927","post","type-post","status-publish","format-standard","hentry","category-education","category-foundations-of-digital-signal-processing","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/posts\/1927","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/comments?post=1927"}],"version-history":[{"count":3,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/posts\/1927\/revisions"}],"predecessor-version":[{"id":2127,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/posts\/1927\/revisions\/2127"}],"wp:attachment":[{"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/media?parent=1927"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/categories?post=1927"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/tags?post=1927"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}