Signal Transmission over Communication Channel

Lesson 03

EE352 Analog Communication Systems

Mansoor Khan

Signal Transmission over a Linear Channel

For an LTI, continuous time system the input output relation is given

by:

where g(t) is the input(transmitted signal) on channel, h(t) is theimpulse response of a LTI channel and y(t) is the convolution outputof g(t) and h(t) as shown above. If

where H(w) is the system transfer function, the from the time convolutionproperty:

Signal Distortion in Transmission

• During transmission the input signal changesfrom g(t) to output signal y(t) as shownbefore.

• The equation below illustrates themodification of g(t):

Where G(w) and Y(w) are the spectra of input

output signals respectively.

• Spectral shaping of input signal by spectral responseof the system is given by(polar form):

• During transmission the spectrum of input signal changes:amplitude changes from |G(w)| to |G(w)||H(w)| andphase changes from θg to θg+θw.

• During transmission some frequency components ofsignal are boosted and some are attenuated therelative phases also vary. In general signal received atthe receiver side is different from transmitted one.

Distortionless Transmission

• Requirement : output waveform be the replica of input signal.

• Characteristics of a distortionless channel:I. The input and output waveform are identical within a multiplicative

constant.

II. A delayed output retaining the input waveform.

Thus in a distortionless channel g(t) and y(t) should satisfy the

condition:

• This shows that for a distortionless channel the amplituderesponse |H(w)| must be constant and phase responseθh(w) must be the linear function of w.

LTI sytem frequency response of a distortionless system. The slope

θh(w) with respect to w is -td, where td is the delay in input signal.If the slope remains constant over the band of interest then eachfrequency component of g(t) undergoes same time delay, but ifamplitude response and phase response vary over the w(freq) scaledifferent frequency components(sinusoids) in g(t) will undergodifferent attenuations and phase delays (varying with the frequncy) asa result the output signal will not be the exact replica of input.

Ideal and Practical Filters

Ideal filters(systems) allows distortionless transmission of input signals allowingcertain band of frequencies and suppressing remaining frequencies. For examplean ideal low pass filter below allows all frequency components below ω = W rad/sto pass through without attenuation and suppress all others above W rad/s. Phaseslope θ(w) is linear (-td) over ω scale which results in a time delay of –td for allfrequency components below W rad/s.Hence for an input signal g(t) bandlimited to W rad/s the output y(t) is g(t) delayedby td.

Since h(t) is the response to the impulse δ(t)

applied at t = 0. The response should be causal

i.e. h(t) = 0 for t < 0.

The impulse response got previously clearly

shows that h(t) ≠ 0 for t < 0, making the filter

noncausal and not realizable.

• Practical filters have gradual characteristics without any discontinuities or jumps as in non causal filters.

• Well known is the butterworth filter for example have the amplitude response of:

The amplitude response approaches ideal for

(order of filter n) n = ∞

A certain tradeoff exists between phase andamplitude response if order of filter is increased the phaseresponse is distorted badly from ideal.

Example

Signal Distortion over a Communication Channel

• Signal distortion in LTI channels are caused by non idealcharacteristics of amplitude or phase response or both. Suppose apulse exist over an interval say (a , b) and is zero elsewhere, thecomponents of its Fourier spectrum phase and magnitude will addin such a way that it is zero elsewhere of interval and exists in (a ,b). For an ideal LTI channel frequency spectrum components aremultiplied by same constant and frequency components delayed bythe same interval resulting in the output replica of input. If for nonidealities frequency components are delayed by different intervalsand attenuated by different amounts due to non ideal phase oramp characteristics the pulse would spread outside the interval.

• This pulse spreading is highly undesirable in TDM systems where itcauses interference with neighboring pulses and consequently withneighboring channel resulting in crosstalk.

Example of non ideal LTI Channel

A low pass filter with transfer function H(ω) shown below is given by:

A pulse g(t) bandlimited to B Hz is applied at the input of this filter find output y(t)

SOLUTION

Distortion caused by Non Linear Channels

Consider an input g and its corresponding output y of a channel related by the

following equation where f() is a nonlinear function:

y = f(g)

Expanding the right handside of above equation by Mclaurin’s series as:

Using the property if the bandwidth of input signal g(t) is B Hz then g^κ is kB

Hz. Hence output y(t) in the above expression will have bandwidth kB Hz, thus

output spectrum extend well beyond the input spectrum giving rise to new

frequency components not contained in the input previously. If the signal is

transmitted over a non linear channel it will cause serious interference

with other signals on the channel (spectral spreading). The problem is more

severe in FDM systems(not in TDM)

Example

Distortion Caused by Multipath Effects

• Arrival of transmitted signal at the receiver by two or more paths with different delays.

• Transmission channel can be modeled as two or more parallel channels over which the same signal travels with varying delay and attenuations.

• For example reflections from hills, buildings and other objects in the path of transmitter and receiver, resulting in direct wave plus reflections arriving at receiver end.

Consider a case of two paths one with the delay of td and unity gain and other with the gain of α and delay of td + Δ. The transfer function of both channels are given by • exp(-jωtd )• α exp(-jω(td + Δ))Overall Transfer function H(ω) of parallel channels is given by:

H(ω) = exp(-jωtd )+α exp(-jω(td + Δ))

Simplifying the above expression:

The magnitude and phase response of H(ω) of the system are periodic in ω with the period of 2*pi/ Δt.The multipath transmission therefore causes non idealities in amplitude and phase characteristics ofchannel resulting in pulse dispersion(spectral spreading) of input signal.

If the gains of both channels are near to each other the received signal undergoes destructiveinterference where the phases are pi radians apart. These frequencies are

multipath null frequencies (n*pi/ Δt) n is odd. For even n there is a constructive

interference and gain is enhanced. Such channels cause frequency selective fading of

transmitted signal.

Fading Channels

• Characteristics of channel vary with time practicallyespecially long range radio communication systems usingionosphere for reflection of transmitted signal the channelcharacteristics vary with the seasonal or weatherconditions heavily.

• Random changes in the propagation characteristics ofchannel.

• Random changes in channel transfer function results inrandom signal attenuation and frequency delay.

• Different frequency components suffer unequalattenuation and phase delays such fading is known asfrequency selective fading. Multipath propagation cancause frequency selective fading.