---
title: "Bandwidth expansion"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Bandwidth_expansion"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T12:04:12.339066+00:00"
instance: "kb-cron"
---

Bandwidth expansion is a technique for widening the bandwidth or the resonances in an LPC filter. This is done by moving all the poles towards the origin by a constant factor 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
. The bandwidth-expanded filter 
  
    
      
        
          A
          ′
        
        (
        z
        )
      
    
    {\displaystyle A'(z)}
  
 can be easily derived from the original filter 
  
    
      
        A
        (
        z
        )
      
    
    {\displaystyle A(z)}
  
 by:

  
    
      
        
          A
          ′
        
        (
        z
        )
        =
        A
        (
        z
        
          /
        
        γ
        )
      
    
    {\displaystyle A'(z)=A(z/\gamma )}
  

Let 
  
    
      
        A
        (
        z
        )
      
    
    {\displaystyle A(z)}
  
 be expressed as:

  
    
      
        A
        (
        z
        )
        =
        
          ∑
          
            k
            =
            0
          
          
            N
          
        
        
          a
          
            k
          
        
        
          z
          
            −
            k
          
        
      
    
    {\displaystyle A(z)=\sum _{k=0}^{N}a_{k}z^{-k}}
  

The bandwidth-expanded filter can be expressed as:

  
    
      
        
          A
          ′
        
        (
        z
        )
        =
        
          ∑
          
            k
            =
            0
          
          
            N
          
        
        
          a
          
            k
          
        
        
          γ
          
            k
          
        
        
          z
          
            −
            k
          
        
      
    
    {\displaystyle A'(z)=\sum _{k=0}^{N}a_{k}\gamma ^{k}z^{-k}}
  

In other words, each coefficient 
  
    
      
        
          a
          
            k
          
        
      
    
    {\displaystyle a_{k}}
  
 in the original filter is simply multiplied by 
  
    
      
        
          γ
          
            k
          
        
      
    
    {\displaystyle \gamma ^{k}}
  
 in the bandwidth-expanded filter. The simplicity of this transformation makes it attractive, especially in CELP coding of speech, where it is often used for the perceptual noise weighting and/or to stabilize the LPC analysis. However, when it comes to stabilizing the LPC analysis, lag windowing is often preferred to bandwidth expansion.


== References ==

P. Kabal, "Ill-Conditioning and Bandwidth Expansion in Linear Prediction of Speech", Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing, pp. I-824-I-827, 2003.