ANALOG COMMUNICATION Lecture 03

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 the impulse response of a LTI channel and y(t) is the convolution output of g(t) and h(t) as shown above. If

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

Signal Distortion in Transmission • During transmission the input signal changes from g(t) to output signal y(t) as shown before. • The equation below illustrates the modification of g(t):

Where G(w) and Y(w) are the spectra of input output signals respectively.

• Spectral shaping of input signal by spectral response of 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)| and phase changes from θg to θg+θw.

• During transmission some frequency components of signal are boosted and some are attenuated the relative phases also vary. In general signal received at the receiver side is different from transmitted one.

Distortionless Transmission • Requirement : output waveform be the replica of input signal. • Characteristics of a distortionless channel: I. II.

The input and output waveform are identical within a multiplicative constant. 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 amplitude response |H(w)| must be constant and phase response θh(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 each frequency component of g(t) undergoes same time delay, but if amplitude response and phase response vary over the w(freq) scale different frequency components(sinusoids) in g(t) will undergo different attenuations and phase delays (varying with the frequncy) as a 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 allowing certain band of frequencies and suppressing remaining frequencies. For example an ideal low pass filter below allows all frequency components below ω = W rad/s to pass through without attenuation and suppress all others above W rad/s. Phase slope θ(w) is linear (-td) over ω scale which results in a time delay of –td for all frequency components below W rad/s. Hence for an input signal g(t) bandlimited to W rad/s the output y(t) is g(t) delayed by 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 and amplitude response if order of filter is increased the phase response is distorted badly from ideal.

Example

Signal Distortion over a Communication Channel • Signal distortion in LTI channels are caused by non ideal characteristics of amplitude or phase response or both. Suppose a pulse exist over an interval say (a , b) and is zero elsewhere, the components of its Fourier spectrum phase and magnitude will add in such a way that it is zero elsewhere of interval and exists in (a , b). For an ideal LTI channel frequency spectrum components are multiplied by same constant and frequency components delayed by the same interval resulting in the output replica of input. If for non idealities frequency components are delayed by different intervals and attenuated by different amounts due to non ideal phase or amp characteristics the pulse would spread outside the interval. • This pulse spreading is highly undesirable in TDM systems where it causes interference with neighboring pulses and consequently with neighboring 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 of channel resulting in pulse dispersion(spectral spreading) of input signal. If the gains of both channels are near to each other the received signal undergoes destructive interference 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 practically especially long range radio communication systems using ionosphere for reflection of transmitted signal the channel characteristics vary with the seasonal or weather conditions heavily. • Random changes in the propagation characteristics of channel. • Random changes in channel transfer function results in random signal attenuation and frequency delay. • Different frequency components suffer unequal attenuation and phase delays such fading is known as frequency selective fading. Multipath propagation can cause frequency selective fading.