A comparison of the classic and relativistic Doppler effect in linear
time-variant channel models
Scott Rickard, Konstantinos Drakakis, Nikolaos Tsakalozos
School of Electrical, Electronic & Mechanical Engineering
University College Dublin, Ireland
Abstract
Consider a receiver, y(t) located initially at a distance
d0 from the origin along the positive x-axis moving
The Doppler effect, the apparent change in the fre- to the right with velocity vy . The position of the
quency of a signal caused by contractions/dilations receiver is described by,
of time when transmitter and receiver have relative
py (t) = vy t + d0
(2)
motion, can be derived using either Newtonian or relativistic mechanics. One may be led to believe that
We define τ to be the time delay such that the
the Newtonian Doppler effect is applicable to sound
waves, while the relativistic Doppler effect is applica- source signal traveling with speed c0 emitted at time
ble to electromagnetic waves, but this is, of course, t − τ reaches the receiver at time t. Clearly, the disnot the case; there are not two Doppler effects. The tance that the signal travels must equal the difference
relativistic model represents a more accurate descrip- between the current receiver position and position of
tion for both acoustic and electromagnetic waves, but the source τ seconds ago. This can be described,
for typical (non-relativistic) speeds in acoustic setc0 τ = |py (t) − px (t − τ )|
(3)
tings, the classic model is sufficient. In this paper, we
derive from first principles both the Newtonian and
= |vy t + d0 − vx (t − τ )|.
(4)
the relativistic input-output relationships for transmitter and receiver moving directly towards or away Solving for τ ,
from each other with constant speed. We compare
the two models and show how the non-relativistic
(vy − vx )t + d0
(5)
,
model can be seen as can be seen as approximation Case 1: py (t) > px (t − τ ), τ =
c0 − vx
of the relativistic one, when velocities are small.
(−vy + vx )t − d0
Case 2: py (t) < px (t − τ ), τ =
(6).
c0 + vx
1
Substituting these back into (4) we obtain (assuming
|vx | < c0 ),
The classic Doppler effect
The treatment in this section is the classic treatment
of the Doppler effect, as opposed to relativistic. Consider a source, x(t) located at the origin moving to
the right with velocity vx . The position of the source
is described by,
px (t) = vx t
(1)
Case 1:
py (t) > px (t),
τ
Case 2:
py (t) < px (t),
τ
(vy − vx )t + d0
,(7)
c 0 − vx
(−vy + vx )t − d0
=
(8)
.
c0 + vx
=
Assuming that the amplitude decays with the recip1
rocal of distance, the input output mapping is,
1
x(t − τ )
(9)
|py (t) − px (t − τ )|

d0
t− c −v

1−v /c
y
0

: py (t) > px (t)
 d0 −(vxx−v0y )t x
c0 −vx
c0 −vy
(10)
=
d0
t+ c +vy

1+v /c0
0

x
:
p
(t)
<
p
(t)
 (vx −vyx)t−d
c0 +vx
y
x
0
y(t) =
c0 +vy
The change in Doppler effect from contraction to expansion occurs when the transmitter and receiver are
collocated (py (t) = px (t)).
2
The relativistic Doppler effect
Figure 1: Space-time diagram for the derivation of
For electromagnetic waves, c0 = c, the speed of light,
the Doppler effect.
and we must account for relativistic effects that we
ignored in the derivation of (10). Consider the same
setup as before, which is now depicted in the spacetime diagram in Figure 1. We have,
can be determined by swapping the signs on the velocities and the original separation distance, as was
O:
(0, 0)
(11) the case in the non-relativistic result. Using (15)
A:
(vx t0 , t0 )
(12) and distance as measured in Minkowski space for the
B:
(d0 , 0)
(13) space-time diagram (for example, see Chapter 14 in
Kleppner and Kolenkow [1973]),
D:
(d, t)
(14)
q
t2A − d2A = 1
(19)
and, as the slope of AD and OC is 1/c0 , the slope of
OA is 1/vx , and the slope of BD is 1/vy ,
we determine the location of point A,
1
!
OA :
t =
d
(15)
vx
vx
c
(dA , tA ) = p
(20)
,p
1
c2 − vx2
c2 − vx2
(16)
OC :
t = d
c0
1
BD :
t = (d − d0 )
(17) Now we can determine the t-intercept in (18),
vy
1 − vx
1
b = q c0
(21)
AD :
t = d+b
(18)
v2
c0
1 − cx2
where b is yet to be determined. Without loss of
generality, we define lx = 1 and we are interested in The intersection of AD and BD yields,
!
lx /l2 = 1/l2 which is the time dilation factor.
bvy
For simplicity, we consider here only the case where
c + d0 cd0 + bc0
(dD , tD ) =
,
v
c0 − vy
1 − cy0
x is located to the left of y. The complementing case
2
(22)
Intersecting OC and BD,
c0 d0
cd0
(dC , tC ) =
,
c0 − vy c0 − vy
Then the length from C to D is,
p
(tD − tC )2 − (dD − dC )2
l2 =
s
2 2
c0 bvy /c
bc0
−
=
c0 − vy
c0 − vy
q
b
=
1 − (vy /c)2
1 − vy /c0
q
v
vx
1 − ( cy )2
1 − c0
=
v · p
1 − cy0
1 − ( vcx )2
When c = c0 ,
s
(23)
l2 =
and defining v =
(24)
(25)
vx
c
vx
c
1+
1−
vy
c
vy
c
(34)
vy −vx
1−vx vy /c2 ,
s
l2 =
1−
1+
1−
1+
v
c
v
c
1 − vc
=q
,
2
1 − vc2
(35)
(26) the input-output relation takes the form,
!

t −

y(t) = x 


(27)
d0
vx −vy
1− v
q c
2
1− vc2





(36)
1− v
q c
v2
The first term in (27) is the same as the scale factor
1− c2
in (10), the second term is the relativistic correction.
The input-output relation takes the form,
Unlike in (10), we see that the Doppler effect elec
tromagnetic transmission depends only on the relat
− d1 .
y(t) = x
(28) tive velocity of transmitter and receiver and not on
l2
the individual velocities.
We solve for d1 by extending AO and BD to their
We have ignored the 1/d amplitude factor. Howpoint of intersection,
ever, the argument used in the classical analysis ap
plies verbatim and thus the amplitude factor is the
d0 vx
d0
(dI , tI ) =
,
(29) same. Note, we could derive the identical factor
vx − vy vx − vy
through the Minkowski diagram, as well.
and note that at that point,
y(tI ) = x(tI /l2 − d1 ) = x(tI )
2.1
Comparison and Approximation
(30)
The three models (classic, relativistic with arbitrary
speed of propagation, and relativistic with speed of
tI = tI /l2 − d1
(31) light propagation) are displayed in Table 1. Note the
lack of symmetry with vx and −vy . That is, the efthus,
fect of x moving toward stationary y with speed v is
different than the effect of y moving toward stationd0
1
d1 = tI (1/l2 − 1) =
−1 .
(32) ary x with speed v. This is not the case when c0 = c.
v x − vy l2
That is, when the speed of signal propagation is the
speed of light, it is impossible to for either x or y to
We state (28) in a similar form as (10) as,
determine whether x is moving toward stationary y

√ vy 2 
1− vcx
1−( c )
or y is moving toward stationary x due to relativistic
d0
0
√
 t − vx −vy 1 − 1− vcy0 · 1−( vcx )2 
effects Kleppner and Kolenkow [1973]. Also note that

√ vy
y(t) = x 


2
1− vcx
vx = v and vx = −v do not shift the frequency by
1−(
)
c
0
· √ vx 2
v
1− cy
1−( c )
the same amount. This difference remains true even
0
(33) when c0 = c.
thus,
3
classic
y(t) =

relativistic
y(t) =
d0
0 −vy
c0 −vx
c0 −vy
v
vy 2
1− x
)
1−(
d0
c0
c
t− vx −vy 1−
vy ·
vx 2
1−
1−(
)
c0
c
v
vy 2
1− x
1−(
)
c0
c
vy ·
v
1−
1−( x )2
c0
c


1−vx /c0
d0 −(vx −vy )t x

1−vx /c0
d0 −(vx −vy )t x 
t− c
√
√
√
√

d
1− v
c
2
1− v2
c
0
r
t− vx −v
y
relativistic (c0 = c)
y(t) =
1−vx /c

d0 −(vx −vy )t x 

1− v
c
r
2
1− v2
c
!







Table 1: A comparison of Doppler effect input-output models.
When |vx | c and |vy | c,
q
v
1 − ( cy )2
p
≈1
1 − ( vcx )2
3
We have derived the input-output model for the
Doppler effect in both a classic and relativistic setting. The relativistic model was shown to give rise to
the classic model when the speeds of the transmitter
and receiver were much less than the speed of light.
(37)
and noting that
1−
1−
1−
vx
c0
vy
c0
=
vx − vy
c0 − vy
Conclusions
(38)
References
and the relativistic model turns into the classic
D. Kleppner and R. Kolenkow. An Introduction to
model.
Mechanics. Mc-Graw-Hill, New York, 1973.
Despite the different forms of the Doppler shift for
source moving toward receiver, receiver moving toward source, and taking into account relativistic effects, we now show that all three of these scenarios
have approximately the same scaling factor. We consider the following three cases where the transmitter
and receiver are approaching one another with speed
v0 : (10) with vx = v0 and vy = 0, (10) with vx = 0
and vy = −v0 , and (35) with v = v0 .
v0
c0
=1−
v0
c0
1
1 + vc00
=1−
v0
c0
+(
v0 2
) − ···
c0
(40)
v0
c0
v0
c0
=1−
v0
c0
1 v0
+ ( )2 − · · ·
2 c0
(41)
1−
s
1−
1+
(39)
As vc00 1, in general, if we ignore second and higher
order terms, we see that all three have time scaling
factor equal to 1 − vc00 .
4