Fading due to Doppler Effect

1. FADING & DOPPLER EFFECT
SWAPNIL BANGERA
ME-EXTC

2. OVERVIEW
• What is Fading?
• Effect of Fading in Wireless Communication
• Doppler Effect (Principle)
• Doppler Effect in Wireless Communication
• Channel Correlation Function and Power Spectra
• Conclusion

3. FADING
• In wireless communication system, a signal experiences multipath propagation which causes rapid
signal level fluctuations in time, called fading.
• Fading effects that characterize radio communication are large scale fading and small scale fading.
• Fading is caused by interference between two or more versions of the transmitted signal which arrive at
the receiver at slightly different times.
• These waves, called multipath waves, combine at the receiver antenna to give a resultant signal which
can vary widely in amplitude and phase, depending on the distribution of the intensity and relative
propagation time of the waves and the bandwidth of the transmitted signals.

4. DOPPLER EFFECT
• The Doppler effect (or Doppler shift) is the change in frequency of a wave (or other periodic event) for
an observer moving relative to its source.
• It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an
observer. Compared to the emitted frequency, the received frequency is higher during the approach,
identical at the instant of passing by, and lower during the recession.
• When the source of the waves is moving toward the observer, each successive wave crest is emitted
from a position closer to the observer than the previous wave. Therefore, each wave takes slightly less
time to reach the observer than the previous wave. Hence, the time between the arrival of successive
wave crests at the observer is reduced, causing an increase in the frequency. While they are travelling,
the distance between successive wave fronts is reduced, so the waves "bunch together".
• Conversely, if the source of waves is moving away from the observer, each wave is emitted from a
position farther from the observer than the previous wave, so the arrival time between successive
waves is increased, reducing the frequency. The distance between successive wave fronts is then
increased, so the waves "spread out".

6. DOPPLER EFFECT IN WIRELESS COMMUNICATION
• The relative motion between the transmitter and the receiver causes Doppler shifts. Local scattering
typically comes from many angles around the mobile. This scenario causes a range of Doppler shifts,
known as the Doppler spectrum. The maximum Doppler shift corresponds to the local scattering
components whose direction exactly opposes the mobile trajectory.
• Due to Doppler spread, fading effects can also be classified as fast fading and slow fading.
• There is always a relative motion between the cell-site transmitter and the mobile receiver. As a result
Doppler effect occurs in the shift of the received carrier frequency. Doppler spectrum is the spectrum of
the fluctuations of the received signal strength.
• Doppler effect results in the inaccurate operation of the system. Proper compensation technique needs
to be implemented to minimise this effect.

7. • Doppler frequency or Doppler shift is given by:
𝑓𝑑 =
𝑉𝑚
𝛌 𝑐
cos 𝜃
• The maximum Doppler frequency will be obtained when the mobile unit is moving in line with the
direction of the received signal, that is, 𝜃 = 0° or cos 𝜃 = 1. The maximum Doppler frequency is given
by:
𝑓𝑑𝑚 =
𝑉𝑚
𝛌 𝑐
• When a pure sinusoidal carrier signal having frequency 𝑓𝑐 is transmitted, the received signal spectrum,
called the Doppler spectrum, will have components in the range 𝑓𝑐 − 𝑓𝑑 to 𝑓𝑐 + 𝑓𝑑, corresponding to
whether the direction of motion of the mobile is away from or towards the direction of the received
signal respectively.
• This simply means that Doppler shift will be positive or negative depending on whether the mobile
receiver is moving toward or away from the base station transmitter.

8. CHANNEL CORRELATION FUNCTION AND POWER
SPECTRA
• Consider ∅ 𝑐(∆𝑓; ∆𝑡) which is known as the spaced-frequency, spaced time correlation function of the
channel.
• Focussing on the time variations of the channel as measured by the parameter ∆𝑡 𝑖𝑛 ∅ 𝑐(∆𝑓; ∆𝑡). The
time variations in the channel are evidenced as a Doppler broadening and, perhaps, in addition as a
Doppler shift of a spectral line.
• In order to relate the Doppler effects to the time variations of the channel, we define the Fourier
transform of ∅ 𝑐 ∆𝑓; ∆𝑡 with respect to the variable ∆𝑡 to be the function 𝑆𝑐 ∆𝑓; 𝛌 .
𝑆𝑐 ∆𝑓; 𝛌 =
−∞
∞
∅ 𝑐(∆𝑓; ∆𝑡)𝑒−𝑗2𝜋𝛌∆𝑡
𝑑∆𝑡
• With ∆𝑓 set to zero and 𝑆𝑐 0; 𝛌 ≡ 𝑆𝑐(𝛌), the relation becomes
𝑆𝑐 𝛌 =
−∞
∞
∅ 𝑐(0; ∆𝑡)𝑒−𝑗2𝜋𝛌∆𝑡
𝑑∆𝑡

9. • The function 𝑆𝑐(𝛌) is a power spectrum that gives the signal intensity as a function of the Doppler
frequency 𝛌. Hence, we call 𝑆𝑐(𝛌) the Doppler power spectrum of the channel.
• From the equation, we observe that if the channel is time invariant, ∅ 𝑐 ∆𝑡 = 1 and 𝑆𝑐(𝛌) becomes
equal to the delta function 𝛿 𝛌 . Therefore, when there are no time variations in the channel, there is
no spectral broadening observed in the transmission of a pure frequency tone.
• The range of values of 𝛌 over which 𝑆𝑐(𝛌) is essentially nonzero is called the Doppler spread 𝐵 𝑑 of the
channel. Since 𝑆𝑐(𝛌) is related to ∅ 𝑐 ∆𝑡 by the Fourier transform, the reciprocal of 𝐵 𝑑 is measure of
the coherence of the channel. That is,
(∆𝑡) 𝑐 ≈
1
𝐵 𝑑
• Clearly, a slow changing channel has a large coherence time or, equivalently, a small Doppler spread.

10. • Now consider the autocorrelation function ∅ 𝑐(𝜏; ∆𝑡) in time domain. Its Fourier transform is given as
𝑆𝑐 𝜏; 𝛌 =
−∞
∞
∅ 𝑐(𝜏; ∆𝑡)𝑒−𝑗2𝜋𝛌∆𝑡
𝑑∆𝑡
• It follows that 𝑆𝑐 𝜏; 𝛌 and 𝑆𝑐 ∆𝑓; 𝛌 are a Fourier transform pair. That is,
𝑆𝑐 𝜏; 𝛌 =
−∞
∞
𝑆𝑐 ∆𝑓; 𝛌 𝑒 𝑗2𝜋𝜏∆𝑓
𝑑∆𝑓
• Furthermore, 𝑆𝑐 𝜏; 𝛌 and ∅ 𝑐(∆𝑓; ∆𝑡) are related by the double Fourier transform
𝑆𝑐 𝜏; 𝛌 =
−∞
∞
−∞
∞
∅ 𝑐(∆𝑓; ∆𝑡)𝑒−𝑗2𝜋𝛌∆𝑡
𝑒 𝑗2𝜋𝜏∆𝑓
𝑑∆𝑡𝑑∆𝑓
• This new function 𝑆𝑐 𝜏; 𝛌 is called the scattering function of the channel. It provides us with a measure
of the average power output of the channel as a function of the time delay and Doppler frequency.

11. CONCLUSION
The intent of this presentation was to discuss the concept of fading in the
context of Doppler effect.