Lecture 15 - EE 104: Introduction to Communications - Winter 2003
Signal Distortion and Equalization. Ideal Filters.
Energy Spectral Density. Autocorrelation.
Lecture Outline
Distortionless Transmission
Distortion Compensation
Ideal Filters
Energy Spectral Density
Autocorrelation
1.
Distortionless Transmission
A system is distortionless if the output equals the input except for an amplitude scaling
and/or delay.
Distortionless transmission in time domain implies y(t) = Kx(t t0) or equivalently h(t) =
KÆ (t t0 ).
Distortionless transmission in frequency domain implies Y (f ) = Ke j2ft0 X (f ) or equivalently H (f ) = Ke j2ft0 (amplitude is constant and phase is linear).
2.
Distortion Compensation
If H (f ) 6= Ke j2ft0 then the system output is a distorted version of the input.
Typical causes of distortion in communication channels are multiple signal reections arriving at the receiver delayed in time.
Many communication systems use an equalizer to compensate for signal distortion caused
by a channel.
The simplest equalizer inverts the channel frequency response H (f ): Heq (f ) = 1=H (f ).
However, this type of equalizer can amplify the noise.
3.
Ideal Filters
An ideal low pass lter passes all positive and negative frequency components of a signal
below a given cuto bandwidth B .
An ideal band pass lter passes all positive and negative frequency components of a signal
between a lower and upper cuto frequency B1 and B2.
4.
Energy Spectral Density (ESD)
Dened as x (f ) = jX (f )j2.
Measures the distribution of signal energy E = R jx(t)j2dt = R x (f )df over frequency.
Properties of ESD include x (f ) 0, x ( f ) = x (f ) for x(t) real, and for x(t) input to a
lter with frequency response H (f ), the lter output y(t) has ESD y (f ) = jH (f )j2x (f ).
5.
Autocorrelation of Energy Signals
Dened for real signals as Rx( ) = R x(t)x(t )dt = x( ) x( ).
Measures the similarity of a signal with a delayed version of itself.
Autocorrelation denes signal energy: E = Rx (0).
Since jRx ( )j Rx (0), can use autocorrelation for signal synchronization.
The autocorrelation is symmetric: Rx ( ) = Rx ( ).
The autocorrelation and ESD are Fourier Transform pairs: Rx ( ) , x (f ).
Main Points:
Most LTI systems, especially communication channels, introduce distortion.
Equalizers compensate for channel distortion, but can enhance the receiver noise.
Energy spectral density measures signal energy distribution across frequency.
Autocorrelation function of an energy signal measures signal self-similarity versus delay.
A signal's autocorrelation and ESD are Fourier transform pairs.