{"id":1884,"date":"2026-08-15T19:49:39","date_gmt":"2026-08-15T19:49:39","guid":{"rendered":"https:\/\/smartdata.ece.ufl.edu\/?p=1884"},"modified":"2026-08-15T19:49:42","modified_gmt":"2026-08-15T19:49:42","slug":"impulse-responses-and-difference-equations","status":"publish","type":"post","link":"https:\/\/smartdata.ece.ufl.edu\/index.php\/2026\/08\/15\/impulse-responses-and-difference-equations\/","title":{"rendered":"Impulse Responses and Difference Equations"},"content":{"rendered":"\n

Convolution and Difference Equations<\/h2>\n\n\n\n

We previously demonstrated that the output of any linear, time-invariant system can be expressed by a convolution of an input and system impulse response. Here, we show that convolutions can be expressed with differences equations. Hence, any linear, time-invariant system can be expressed by difference equations.<\/p>\n\n\n\n

The Difference Equation Form of Convolution<\/h3>\n\n\n\n

If we use the commutativity property (\"x[n] * h[n] = h[n] * x[n]\"), we see that the convolution can be written as

  <\/span>   <\/span>\"\[y[n] = \sum_{m=-\infty}^{\infty} x[m] h[n - m] = \sum_{m=-\infty}^{\infty} h[m] x[n - m]\]\"<\/p> Hence, we can write this as:

  <\/span>   <\/span>\"\[y[n] = \cdots + h[-2] x[n+2] + h[-1] x[n+1] + h[0] x[n-0] + h[1] x[n-1] + h[2] x[n-2] + \cdots\]\"<\/p> Since \"\ldots,h[-2],h[-1],h[0],\ldots\" are all constant values, the expression above is a difference equation.<\/p>\n\n\n\n

Impulse Response Properties<\/h3>\n\n\n\n

Using our new definition, we can we can derive relationship between our impulse response \"h[n]\" and system properties.<\/p>\n\n\n\n

Memoryless<\/h3>\n\n\n\n

An LTI system with impulse response \"h[n]\" is memoryless if

  <\/span>   <\/span>\"\[h[n] = a \, \delta[n]\]\"<\/p> for any scalar constant \"a\".<\/p>\n\n\n\n

Causal \/ Anti-Causal<\/h3>\n\n\n\n

An LTI system with impulse response \"h[n]\" is causal if

  <\/span>   <\/span>\"\[h[n] = 0 \qquad \textrm{for } n < 0\]\"<\/p><\/p>\n\n\n\n

An LTI system with impulse response \"h[n]\" is anti-causal if

  <\/span>   <\/span>\"\[h[n] = 0 \qquad \textrm{for } n \geq 0\]\"<\/p><\/summary>\n

A causal system must be dependent on only past and present values. As a difference equation, this can be written as

  <\/span>   <\/span>\"\[y[n] = h[0] x[n-0] + h[1] x[n-1] + h[2] x[n-2] + \cdots\]\"<\/p><\/p>\n\n\n\n

Hence, this is equivalent to stating \"h[n] = 0\" for all \"n < 0\":

  <\/span>   <\/span>\"\[y[n] = \cdots + (0) x[n+2] + (0) x[n+1] + h[0] x[n-0] + h[1] x[n-1] + h[2] x[n-2] + \cdots\]\"<\/p><\/p>\n<\/details>\n\n\n\n

BIBO Stable<\/h3>\n\n\n\n

An LTI system with impulse response \"h[n]\" is BIBO stable if

  <\/span>   <\/span>\"\[\sum_{n=-\infty}^{\infty} |h[n]| < \infty\]\"<\/p><\/p>\n\n\n\n

Finite \/ Infinite Impulse Responses<\/h3>\n\n\n\n

FIR Filters<\/h4>\n\n\n\n

If the impulse response \"h[n]\" is finite in length (i.e., the number of non-zero values is finite), then the number of terms in the difference equation is finite. We refer to these types of linear, time-invariant systems as finite impulse response filters.<\/p>\n\n\n\n

IIR Filters<\/h4>\n\n\n\n

If the impulse response \"h[n]\" is infinite in length, the linear, time-invariant systems is a infinite impulse response filter.<\/p>\n\n\n\n

Relative to the difference equation, the impulse response \"h[n]\" may be infinite to two reasons.<\/p>\n\n\n\n

The number of difference equation terms is infinite.<\/p>\n\n\n\n

The difference equation has recursion (i.e., \"y[n]\" is a function of \"y[n-N]\" for any \"N\")<\/p>\n\n\n\n

When we have recursive terms, the difference equation can be written as

  <\/span>   <\/span>\"\[\cdots + g[-2] y[n+2] + g[-1] y[n+1] + g[0] y[n-0] + g[1] y[n-1] + g[2] y[n-2] + \cdots \ = \ \cdots + h[-2] x[n+2] + h[-1] x[n+1] + h[0] x[n-0] + h[1] x[n-1] + h[2] x[n-2] + \cdots\]\"<\/p> This can be written as convolutions

  <\/span>   <\/span>\"\[g[n] * y[n] = h[n] * x[n]\]\"<\/p> Hence, this is the general expression for relating any difference equation to convolutions.<\/p>\n\n\n\n

Graphically understanding difference equations<\/h2>\n\n\n\n

Convolution can be seen as a graphical process:<\/p>\n\n\n\n

Plot \"x[n-m]\" with dependent variable \"n\" (\"m\" shifts the signal).<\/p>\n\n\n\n

Plot \"h[n]\" with dependent variable \"n\".<\/p>\n\n\n\n

For each shift, compute \"h[m] x[n-m]\". These are just weighted version of \"x[n-m]\".<\/p>\n\n\n\n

Sum your elements to get \"y[n] = \sum_{m=-\infty}^{\infty} h[m] x[n-m]\".<\/p>\n\n\n\n

Graphical Difference Equation Examples<\/p>\n\n\n\n

Below are a collection of graphical examples of discrete-time convolution \/ difference equation solutions.<\/p>\n\n\n\n

Box and an impulse<\/h3>\n\n\n\n
\n
\n

\"x[n] = u[n] - u[n-3]\"
\"h[n] = \delta[n - 2]\"
\"y[n] = x[n] * h[n]\"<\/p>\n<\/div>\n\n\n\n

\n
\"An
An animation of a convolution between a box and a shifted impulse <\/figcaption><\/figure>\n<\/div>\n<\/div>\n\n\n\n

Two boxes<\/h3>\n\n\n\n
\n
\n

\"x[n] = u[n] - u[n-3]\"
\"h[n] = u[n] - u[n-3]\"
\"y[n] = x[n] * h[n]\"<\/p>\n<\/div>\n\n\n\n

\n
\"An
An animation of a convolution between two boxes <\/figcaption><\/figure>\n<\/div>\n<\/div>\n\n\n\n

Box and exponential<\/h3>\n\n\n\n
\n
\n

\"x[n] = u[n] - u[n-3]\"
\"h[n] = e^{(-1/2) n} u[n]\"
\"y[n] = x[n] * h[n]\"<\/p>\n<\/div>\n\n\n\n

\n
\"An
An animation of a convolution between a box and an exponential <\/figcaption><\/figure>\n<\/div>\n<\/div>\n\n\n\n

Two exponentials<\/h3>\n\n\n\n
\n
\n

\"x[n] = e^{(-1/2) n} u[n]\"
\"h[n] = e^{(-1/2) n} u[n]\"
\"y[n] = x[n] * h[n]\"<\/p>\n<\/div>\n\n\n\n

\n
\"An
An animation of a convolution between two exponentials<\/figcaption><\/figure>\n<\/div>\n<\/div>\n\n\n\n

Finite Ramp and Box<\/h3>\n\n\n\n
\n
\n

\"x[n] = \delta[n-1] + 2 \delta[n-2]\"
\"h[n] = u[n] - u[n-3]\"
\"y[n] = x[n] * h[n]\"<\/p>\n<\/div>\n\n\n\n

\n
\"An
An animation of a convolution between a ramp and a box<\/figcaption><\/figure>\n<\/div>\n<\/div>\n\n\n\n

Random Numbers and Box<\/h3>\n\n\n\n
\n
\n

\"x[n] =\" random signal
\"h[n] = u[n] - u[n-3]\"
\"y[n] = x[n] * h[n]\"<\/p>\n<\/div>\n\n\n\n

\n
\"An
An animation of a convolution between a random signal and a box<\/figcaption><\/figure>\n<\/div>\n<\/div>\n\n\n\n

Step function and an exponential<\/h3>\n\n\n\n
\n
\n

\"x[n] = u[n]\"
\"h[n] = e^{(-1/2) n} u[n]\"
\"y[n] = x[n] * h[n]\"<\/p>\n<\/div>\n\n\n\n

\n
\"An
An animation of a convolution between a step and an exponential<\/figcaption><\/figure>\n<\/div>\n<\/div>\n\n\n\n

Step function and a box<\/h3>\n\n\n\n
\n
\n

\"x[n] = u[n]\"
\"h[n] = u[n] - u[n-2]\"
\"y[n] = x[n] * h[n]\"<\/p>\n<\/div>\n\n\n\n

\n
\"An
An animation of a convolution between a step and a box<\/figcaption><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Convolution and Difference Equations We previously demonstrated that the output of any linear, time-invariant system can be expressed by a convolution of […]<\/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-1884","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\/1884","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=1884"}],"version-history":[{"count":8,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/posts\/1884\/revisions"}],"predecessor-version":[{"id":1919,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/posts\/1884\/revisions\/1919"}],"wp:attachment":[{"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/media?parent=1884"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/categories?post=1884"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/smartdata.ece.ufl.edu\/index.php\/wp-json\/wp\/v2\/tags?post=1884"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}