# Connexions

You are here: Home » Content » Intersymbol Interference

### Recently Viewed

This feature requires Javascript to be enabled.

# Intersymbol Interference

Module by: Don Johnson. E-mail the author

One drawback of the general strategy employed here is that the bit interval has been lengthened to accommodate channel dispersion. In many cases, we cannot accept such limitations, or the transmitter cannot do anything about the dispersion. Consequently, there is "spillover" of what occurred in one bit interval into another.

For example, let the transmitted signal set, defined over the interval 0t<T 0 t T , be s 0 t=0 s 0 t 0 s 1 t=ET s 1 t E T and the channel be characterized by h CH tτ=e(a)(tτ)utτ h CH t τ a t τ u t τ . A typical received signal would then resemble that shown in Figure 1. The effect of what happens in other bit intervals affecting the present interval is termed intersymbol interference (ISI). Many techniques have been developed to combat ISI; we explore only one here.

Let the signal set be an antipodal signal set. The entire transmitted signal is given by: st= k = B k s 0 tkT s t k B k s 0 t k T where B k =±1 B k ± 1 according to the value of the k th k th bit. The signal s 0 t s 0 t is defined over 0 T 0 T only. The received waveform is given by rt=t h CH tτsτd τ +nt r t τ t h CH t τ s τ n t where nt n t is additive, white Gaussian noise. The received signal can be rewritten as rt= k = B k s 0 * tkT+nt r t k B k s 0 * t k T n t where s 0 * t=0T h CH tτ s 0 τd τ s 0 * t τ 0 T h CH t τ s 0 τ . The presumption now is that the duration of s 0 * t s 0 * t is longer than TT. Let us further assume that the receiver has a front end containing a matched filter having impulse response h f tτ h f t τ , the output of which is sampled every t=lt t l t to produce the sequence r ~ l T r ~ l T .

r ~ l T =rt* h f tτ|t=lT= k =l1 B k nT(n+1)T h f lTτ s 0 * τnTd τ +lT h f lTτnτd τ r ~ l T t l T r t h f t τ k l 1 B k τ n T n 1 T h f l T τ s 0 * τ n T τ l T h f l T τ n τ
(1)
r ~ l T = k =l1 B k s ~ lTkT+ n ~ l T r ~ l T k l 1 B k s ~ l T k T n ~ l T If there were no intersymbol interference, s ~ lTkT=0 s ~ l T k T 0 , kl k l , and we would compare this sequence with zero to determine optimally what was transmitted. When intersymbol interference is present, we can usually do better than that. Let s ~ l = s ~ lT s ~ l s ~ l T and n ~ l = n ~ lT n ~ l n ~ l T be the sample values of the signal and noise, respectively. We have r ~ l = k = B k s ~ l - k + n ~ l r ~ l k B k s ~ l - k n ~ l . As s ~ l =0 s ~ l 0 , l<0 l 0 , we have r ~ l = k =l B k s ~ l - k + n ~ l r ~ l k l B k s ~ l - k n ~ l . The sum is a convolution sum; thus the transmitted sequence B l B l is filtered by a discrete-time filter having unit sample response s ~ l s ~ l .

One method of "recovering" B l B l would be to find the best inverse filter to remove the ISI. Another method is to use previous determinations of B l B l to remove the interference. Let B ^ l B ^ l denote the receiver's version of the transmitted sequence B l B l . Consider subtracting from the filtered and sampled received signal the terms corresponding to previous bit determinations. In essence, we are feeding-back previous decisions to help determine what the present bit might be. r ~ l k =l1 B ^ k s ~ l - k = B l + k =l1( B k B ^ k ) s ~ l - k + n ~ l r ~ l k l 1 B ^ k s ~ l - k B l k l 1 B k B ^ k s ~ l - k n ~ l (Note that we have normalized s ~ l s ~ l so that s ~ 0 =1 s ~ 0 1 ). Obviously, if the receiver does not make an error ( B ^ l = B l B ^ l B l ), this quantity equals B l + n ~ l B l n ~ l , the simple signal-in-noise situation, and we are left with the digital transmission scheme without ISI but with the unavoidable additive noise. One usually only "feedbacks" those B ~ l B ~ l for which s ~ l s ~ l is significant. This method is termed decision-feedback equalization.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

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

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?

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