Another wireless phenomenon that can affect system performance is Doppler Shift. In practice, infor- mation is often sent in a radio system under mobile conditions. The impacts of this can be considerable, especially in satellite and railway communications. Consider a sine wave being transmitted via a radio link in a mobile environment, where a transmitter is moving at velocity Vm at an instantaneous distance d0
from a stationary receiver.
As the transmitter moves towards the receiver, the instantaneous distance d0 will shrink, impacting
transmission time as a function of time [1]:
τ(t) =τ0− Vm
c t, (2.19)
where c is the speed of light in free-space, 3·108 m/s, and τ0 = d0/c is the starting transmission time.
Given this, the transmitted sine wave can be formulated by Euler’s formula as [1]:
r(t) =Arej2πfc[t−τ(t)]=Arej[2π(fc+fd)t−φ], (2.20)
where fc is the sine waves carrier frequency, Ar is the amplitude of the received signal, φ = 2πfctau0 is
the current phase offset, andfd is the Doppler shift caused by movement:
fd=
Vm
c fccos(θ), (2.21)
where θ is the direction of movement, for zero degrees being moving straight towards the receiver. The frequency shift observed by the receiver is positive or negative depending on the direction of movement, where magnitude is maximized by the cosine when moving exactly toward or away from the transmitter. Commonly these maximal outcomes of (2.21) are described as the maximum Doppler shift fM =±Vcmfc
of bandwidthBD = 2fM.
However, as shown in Section 2.2, rarely is there one ray (see Figure 2.2) in a wireless channel. Each ray is affected differently, and as consequence the frequency domain representation of the signal is affected by a Doppler spectrum, or Doppler spread D(λ). Consider the wireless channel responding to probing
impulse responsesδ(t) at time delays τ in the form h(τ, t), where for the inputx(t), the channel output is y(t) =x(t)~h(t): h(τ, t) = L X i=1 βiejφiδ(t−τi). (2.22)
The autocorrelation of the observed impulse response at two different delays and times can then be formu- lated as [1]:
Rhh(τ1, τ2;t1, t2) =Rhh(τ1; ∆t)δ(τ1−τ2), (2.23)
for ∆t=t2−t1. GivenRhh, the autocorrelation in the frequency domain given the time-varying frequency
domain impulse responseH(f;t) =R−∞∞ h(τ;t)e−jωτdτ is formulated in [1] as:
RHh(f1, f2; ∆t) =RHh(∆f; ∆t), (2.24)
where ∆f = f2 −f1. The channel is assumed to have Wide-Sense Stationary Uncorrelated Scattering
(WSSUS), which is valid for most wireless channels [1]. A WSSUS wireless channel is one in which the delays τi are uncorrelated, and the correlations between paths of equal delays are stationary, or time-
invariant. Given this, the autocorrelation can be estimated asRHh(∆f) if the channel is slowly time-varying
or time-invariant.
Finally, we can derive the Doppler spectrumD(λ) using the Fourier transform of (2.24):
RHH(∆f;λ) = Z ∞ −∞ RHh(∆f; ∆t)e−j2πλ∆td(∆t), (2.25) such that: D(λ) =RHH(0;λ). (2.26)
The spectrum is limited by±fM, and the amount of variation of the spectrum over frequency is described
by the Doppler spread:
BD,rms2 = RfM −fMλ 2D(λ)dλ RfM −fMD(λ)dλ . (2.27)
Figure 2.7: A flow chart adapted from [1] summarizing equations (2.23) through (2.25). Arrows indicate Fourier (down) and inverse Fourier (up) transforms.
In the most basic multi-path case, (2.26) takes the form:
D(f) = 1 2πfm 1− f fm 2−1/2 ,|f| ≤fm, (2.28)
or Jakes Doppler spectrum, where (2.21) is assumed and can be plotted over normalized frequency as seen in Figure 2.8.
0 0.005 0.01 0.015 0.02 0.025 0.03 Amplitude
Taps as Beacon Passes Sensor
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Tau (seconds) 10-8
(a) Five taps plotted to display phasor delay and magnitude.
-6 -4 -2 0 2 4 6 Frequency w (Hz) 10-10 10-8 10-6 10-4 10-2 100 D (dB) Doppler Spectrum
(b) Jakes Doppler spectrum (b) of the scenario described by Figure2.4.
Figure 2.8: Taps (a) (See Appendix A.1) calculated from the phasers and time delays described by Fig- ure 2.4 as the transmitting car passes by the receiver at x = 25m. Jakes Doppler spectrum (b) where frequency offset is maximal at ±fM (2.21), or movement directly away from and towards the receiver.
Small fluctuations in the spectrum is caused by movement within the channel caused by the leading and lagging vehicles.
However, there is a whole field dedicated to modeling unique cases of Doppler shift. Some common cases in addition to Figure2.8are displayed in Figure 4.9 of [1]. The series of experiments include a series of impulse responses (2.22) and their Fourier transforms, describing a LOS experiment using a stationary radio transmit-receive pair in a stationary environment (BD = 0Hz see (2.27)), and a LOS experiment
using a stationary receiver, but mobile transmitter that moved randomly within a 12 meter radius of a fixed point to simulate a pseudo-stationary mobile user pacing on their telephone (BD = 4.9Hz). An obstructed
LOS (OLOS) experiment is described, using stationary devices 4 meters apart, but with heavy pedestrian traffic around the transmitter (BD = 5.7Hz), and an OLOS experiment with stationary devices, but the
transmitter is rotated at a rate of 2.5 rotations per second (BD = 5.2Hz). Each experiment describes a
family of movement, corresponding to time and frequency domain Doppler signals that can be expected in each, while also showing that BD can be equal for very different spectrum’s D(λ) shapes.