# Input Output Models of Wireless Channels

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```					6.450 Introduction to Digital Communication                                      November 27, 2002
MIT, Fall 2002

Lecture 21: Input/output models for wireless

1    Input/Output Models of Wireless Channels
Suppose a transmitting antenna sends a sinusoid, cos(2πf t), which is received at a re-
ceiving antenna after reﬂection from some intermediate object. The response will be
a function of each antenna pattern and of the intermediate object’s reﬂection pattern.
In addition, there will be an attenuation factor that is a function of the distance from
transmitting antenna to reﬂector and from reﬂector to receiving antenna. To describe
this path in terms of an input/output relationship between transmitter and receiver, we
simply multiply all of these attenuation terms together as a single attenuation factor βj (t)
at time t from transmitter to receiver via a given path j. For the example of a perfectly
reﬂecting wall in lecture 20, then,
|α|                             |α|
β1 (t) =                      β2 (t) =                                 (1)
r0 + vt                       2d − r0 − vt
where the ﬁrst expression is for the direct path and the second for the reﬂected path.
Similarly, we deﬁne τj (t) as the propagation delay on path j from transmitter to receiver.
Thus, for the reﬂecting wall example,
r0 + vt ∠φ1 (f )                           2d − r0 − vt ∠φ2 (f )
τ1 (t) =          −                        τ2 (t) =               −                 (2)
c     2πf                                    c        2πf
The term ∠φj (f ) here is to account for possible phase changes at the transmitter, reﬂector,
and receiver. For the example here, there is a phase reversal at the reﬂector so we can take
φ1 = 0 and φ2 = π. With these deﬁnitions, the response to a sinusoid for the reﬂecting
wall example can be expressed as

Er (f, t) =    [β1 (t) exp{i2πf [t − τ1 (t)]}] +     [β2 (t) exp{i2πf [t − τ2 (t)]}]   (3)

For an arbitrary number k of paths, this expression becomes
k
Er (f, t) =          [βj (t) exp{i2πf [t − τj (t)]}]                   (4)
j=1

In the previous lecture, our focus was on the electromagnetic eﬀects which give rise to time-
varying attenuation and path delay (along with the very notion of multiple propagation
paths). Today, we abstract from these electromagnetic eﬀects to study their eﬀect on
communication. The attenuations and path delays are now taken as given and we want to
ﬁnd an input/output characterization of the channel. We will use the physical mechanisms

1
to get some order-of-magnitude sense of how the parameters vary with time, but otherwise,
we simply explore the consequences of the assumed sinusoidal response in (4).
The eﬀect of the Doppler shift is not immediately evident in (4). Recall that the Doppler
shift Dj on path j is deﬁned as −f vj /c where vj is the velocity at which the path length
is changing. Thus we can express τj (t) in (2) as

Dj t
τj (t) = τj −                                         (5)
f

Here τj is assumed to be constant with respect to both t and f . Dj is linearly increasing
in f , so that Dj /f is not a function of f , and thus τj (t) is also independent of f . The
attenuations in (4) are usually slowly varying functions of frequency. These variations
follow from the time-varying path lengths (as in (1)) and also from frequency dependent
antenna gains. For bands that are narrow relative to the carrier frequency, we can safely
omit this frequency dependence. As we see later, however, there is an important frequency
dependence in (4) that arises from multiple paths at diﬀerent delays and Doppler shifts.
The behavior of (4) does not depend critically on the number of paths, k, so this will be
suppressed from now on.

1.1    Time-varying System Functions
We now derive a time-varying system function and then a time-varying impulse response
for the above channel; the procedure is quite similar to that for linear time-invariant (LTI)
channels. View (4) as giving the response to sinusoids at arbitrary frequencies (within
ˆ
the band of interest). Deﬁne the time-varying system function h(f, t) as

ˆ
h(f, t) =        βj (t) exp{i2πf τj (t)}                       (6)
j

Substituting this in (4), we see that the response to an input cos(2πf t) is
ˆ
[h(f, t) exp(i2πf t). More generally, the response to an input cos(2πf t + φ) is
ˆ
h(f, t) exp(i2πf t + φ) . As usual with system functions, it is convenient to deﬁne
ˆ                                   ˆ           ˆ
h(f, t) for negative frequencies as h(−f, t) = h∗ (f, t). We can then view the response to
ˆ
an input exp(i2πf t) as h(f, t) exp(i2πf t) for both f > 0 and f < 0. Using linearity, the
response to a weighted sum of sinusoids, say x(t) = k xk exp{i2πfk t} is

y(t) =          ˆ
xk h(fk , t) exp{i2πfk t}                         (7)
k

We can represent any L2 input x(t) (in the frequency band of interest) by a Fourier
transform
∞                                             ∞
x(t) =        ˆ
x(f ) exp(i2πf t)df     ;       ˆ
x(f ) =        x(t) exp(−i2πf t)dt
−∞                                             −∞

2
Using linearity on a continuum of sinusoids in the same way as on the sum in (7), the
∞
ˆ
response to x(t) = −∞ x(f ) exp(i2πf t)df is
∞
y(t) =        ˆ ˆ
x(f )h(f, t) exp(i2πf t)df                             (8)
−∞

There is a temptation here to blindly imitate the theory of LTI linear systems and to
ˆ           ˆ ˆ
confuse the Fourier transform of y(t), namely y (f ), with x(f )h(f, t). This is wrong math-
ˆ t) is a non-constant function of t; this dependence on t prevents
ematically whenever h(f,
taking the Fourier transform of (8) in any straightforward way.
ˆ           ˆ ˆ
Confusing y (f ), with x(f )h(f, t) is also wrong physically. The response, for a given f , to
ˆ                    ˆ ˆ
x(f ) exp(i2πf t) is x(f )h(f, t) exp(i2πf t). This is a narrow band waveform rather than a
ˆ
sinusoid because of Doppler shifts. This means that y (f ) at a given f also depends on
ˆ
x(f ) over a range of f .
ˆ           ˆ ˆ                                  ˆ
Finally, confusing y (f ), with x(f )h(f, t) is non-sensical, because y (f ) is not a function of
ˆ t)ˆ(f ) is.
t and h(f, x

1.2     The impulse response and the convolution equation
Fortunately, (8) can still be used to derive a very satisfactory form of impulse response
and convolution equation. Deﬁne the time-varying impulse response h(τ, t) as the inverse
ˆ
Fourier transform1 (in τ ) of h(f, t), where t is viewed as a parameter. In particular,
∞                                               ∞
h(τ, t) =        ˆ
h(f, t) exp(i2πf τ )df          ˆ
h(f, t) =        h(τ, t) exp(−i2πf τ )dτ    (9)
−∞                                               −∞

ˆ
Intuitively, we regard h(f, t) as a system function that is slowly changing with t, and view
h(τ, t) as a channel ﬁlter whose impulse response (as a function of τ ) is slowly changing
with t. If we substitute the second part of (9) into (8), we get
∞       ∞

y(t) =                  x(f )h(τ, t) exp[i2πf (t − τ )]dτ df
ˆ
f =−∞ τ =−∞

Interchanging the order of integration and recognizing the integration over f as the inverse
ˆ
Fourier transform of x(f ), we get the convolution equation for time-varying ﬁlters,
∞
y(t) =        x(t − τ )h(τ, t)dτ                               (10)
−∞

This expression is really quite nice. It says that the eﬀect of mobile cell phones, arbitrarily
moving reﬂectors and absorbers, and all of the complexities of solving Maxwell’s equations,
ﬁnally reduce to an input/output relation between transmit and receive antennas which
1             ˆ
The function h(f, t) as described in (6) is not an L2 function of f and thus does not have an inverse
Fourier transform in the usual sense. We discuss this later.

3
is simply represented as the impulse response of a linear time-varying channel ﬁlter. That
is, h(τ, t) is the response at time t to an impulse at time t − τ . If h(τ, t) is a constant
function of t, then this is the conventional LTI impulse response.
ˆ
For the particular form of h(f, t) in (6), the inverse transform h(τ, t) is

h(τ, t) =           βj (t)δ{τ − τj (t)}                   (11)
j

where δ is the Dirac impulse function. These idealized, non-physical, impulses arise here
because of our earlier assumption that βj (t) and τj (t) are not functions of frequency, which
we justiﬁed by our interest only in inputs over a narrow band of frequencies around some
carrier fc . Physically, these delta functions arose from viewing reﬂectors solely through
the ray tracing approximation and by ignoring the frequency attenuation of the antennas.
ˆ
We can see in (6) that if x(f ) is limited to a given band, then it makes no diﬀerence what
ˆ t) is outside of that band. In the same way, if the impulses in (11) were ﬁltered to
h(f,
eliminate the out-of-band components, the response to a band-limited input would remain
the same. To see this more clearly, we can substitute (11) into (10), getting

y(t) =             βj (t)x(t − τj (t))                   (12)
j

Note that if δ{τ − τj (t)} in (11) were replaced with a sinc function centered on τj (t) with
a bandwidth wider than than of x(t), then the response in (12) would not be changed.
Perhaps more to the point, if we used a more elaborate electromagnetic model, the re-
sponse from the jth path would be a linear time-varying ﬁlter in its own right, so that
the overall response would again be a linear time-varying ﬁlter.

2     Baseband equivalent system functions and impulse
responses
Our next step in being able to interpret these time-varying ﬁlters is to represent the above
bandpass functions in terms of baseband equivalents. Recall that for any complex base-
band waveform u(t) of nominal bandwidth W/2, the bandpass equivalent real waveform
x(t) (of nominal bandwidth W around fc ) is given by

x(t) = u(t)ei2πfc t + u∗ (t)e−i2πfc t

In transform terms, x(f ) = u(f − fc ) + u∗ (−f + fc ). The positive bandwidth part of x(t)
ˆ      ˆ           ˆ
is simply u(t) shifted up by fc . As we saw before, this relationship can be expressed as

x(f + fc ) ; f > −fc
ˆ
ˆ
u(f ) =                                                     (13)
0      ; f ≤ −fc
Earlier, we viewed the baseband waveform as being modulated up to bandpass, then
combined with noise at bandpass, and then demodulated back down to baseband.

4
- Discrete                 u(t)                     x(t)
-   Modulation
Encoder                                                     ?
Channel
Binary                                                       ﬁlter
Interface                                                    h(τ, t)

⊕WGN
N (t)
              Discrete         v(t) Demodulation y(t)
Decoder

Figure 1. Modulation and channel ﬁltering for wireless communication.

Here, the bandpass waveform x(t) is ﬁltered by the linear-time-varying ﬁlter h(τ, t) before
the addition of noise and demodulation (see Figure 1).
The received bandpass waveform y(t) given by (10) is then demodulated into the baseband
waveform v(t). We neglect the addition of noise for the time being. In terms of the
frequency domain, this demodulation is deﬁned by
y (f + fc ) ; f > −fc
ˆ
ˆ
v (f ) =                                                           (14)
0      ; f ≤ −fc
We next show that it is possible to avoid the tedium of converting the baseband input
u(t) to passband, convolving with h(τ, t), and then converting the received waveform y(t)
back down to baseband. We do this by deﬁning the baseband equivalent of h(τ, t). This
baseband equivalent (in transform form) is deﬁned by
ˆ
h(f + fc , t) ; f > −fc
ˆ
g (f, t) =                                                              (15)
0         ; f ≤ −fc
The baseband equivalent g(τ, t) is then deﬁned by the Fourier relations
∞                                                   ∞
g(τ, t) =           ˆ
g (f, t) exp(i2πf τ )df            ˆ
g (f, t) =        g(τ, t) exp(−i2πf τ )dτ   (16)
−∞                                                      −∞

ˆ
We see that if a baseband sinusoid at frequency f , say u(f ) exp{i2πf t}, is to be transmit-
ˆ
ted, it is translated up to the positive frequency passband sinusoid u(f ) exp{i2π(f +fc )t}.
The positive frequency passband response is then
ˆ ˆ                                   ˆ g
u(f )h(f +fc , t) exp{i2π(f +fc )t} = u(f )ˆ(f, t) exp{i2π(f +fc )t
ˆ g
Finally, the baseband response is u(f )ˆ(f, t) exp{i2πf t}. We can now take an arbitrary
∞
ˆ
baseband input u(t) = −∞ u(f ) exp{i2πf t}df and use linearity on the above response to
a sinusoid to get
∞
v(t) =              ˆ g
u(f )ˆ(f, t) exp(i2πf t)df                         (17)
−∞

5
ˆ                    ˆ g                  ˆ
As with bandpass functions, v (f ) is not equal to u(f )ˆ(f, t) (unless g (f, t) is constant
in t). However, we can derive the convolution equation for v(t) as before. Speciﬁcally,
substituting the second part of (16) into (17),
∞             ∞

v(t) =                        u(f )g(τ, t) exp[i2πf (t − τ )]dτ df
ˆ
f =−∞ τ =−∞

ˆ
Recognizing the integration over f as the inverse transform of u(f ), we get the time-
varying low pass convolution equation,
∞
v(t) =               u(t − τ )g(τ, t)dτ                  (18)
−∞

ˆ
It is helpful in interpreting this to look at the particular form of h(f, t) in (6). At baseband,
using (15), we have

ˆ
g (f, t) =             βj (t) exp{−i2π(f + fc )τj (t)}            (19)
j

Taking the inverse transform with respect to f ,

g(τ, t) =            βj (t) exp{−i2πfc τj (t)} δ[τ − τj (t)]         (20)
j

Substituting into (18),

v(t) =             βj (t) exp{−i2πfc τj (t)} u[t − τj (t)]          (21)
j

This, ﬁnally, can be interpreted in the time domain. The baseband output is the sum,
over each path, of the delayed replicas of the baseband input. The magnitude of the j th
such term is the magnitude of the response on the given path; this changes slowly, with
signiﬁcant changes occuring on the order of seconds or more. The phase exp{i2πfc τj (t)}
typically varies several orders of magnitude faster than the magnitude. Writing out the
delay τj (t) in terms of the Doppler shift from (5), we get

v(t) =        βj (t) exp{−i2π(fc τj − Dj t) u[t − τj (t)]              (22)
j

Now recall that at the receiver, the carrier frequency is recovered as the data is detected.
Thus if there is only one path with a Doppler shift D, the recovered carrier will be at
fc − D. When there are multiple paths, the recovered carrier will be altered by some
sort of average between the diﬀerent Doppler shifts. Thus what is important is not the
Doppler shifts themselves, but rather the spread between them, which is what determines
how far they are from this average formed in the carrier recovery.

6
We can express this by modifying (22) by changing the carrier frequency fc to the recovered
carrier frequency fc and changing each Doppler shift Dj to the Doppler shift Dj relative
to the recovered carrier, yielding

v(t) =        βj (t) exp{−i2π(fc τj − Dj t)} u[t − τj (t)]       (23)
j

For each path, the term exp{−i2π(fc τj − Dj t)} can be viewed as a time-varying phase
angle. The term fc τj is simply a constant, but Dj t changes linearly with time; when
t changes by 1/(4Dj ), the phase on the path changes by π/2 and the real part of the
response becomes imaginary and the imaginary part becomes real. In other words, the
jth component of the baseband channel response h(τ, t) changes signiﬁcantly over the time
interval 1/(4Dj ). Deﬁne the Doppler spread on the channel as the diﬀerence between the
largest and the smallest Doppler shift over signiﬁcant paths. This deﬁnition is imprecise
since we are trying to capture the set of diﬀerent Doppler shifts in one order-of -magnitude
expression. Since the recovered carrier has been shifted by an approximate average over
the Doppler shifts, the modiﬁed Doppler shifts Dj range from about −D/2 to +D/2. It
follows that the interval of time over which the paths change signiﬁcantly is on the order
of 1/(2D). This interval is called the coherence time Tc of the channel,
1
Tc =                           (24)
2D
Neither D nor Tc are exact quantities; their purpose is to provide a guideline for how
quickly the channel is changing, and to see that this time is inversely related to the
We have now derived a continuous time baseband model for wireless communication.
Any physical reﬂectors, shadowing, scattering, etc. can all be modeled in this way as
time-varying delayed responses to the input.

2.1    A Discrete Time Baseband Model
The ﬁnal step in creating a useful channel model is to convert the continuous time channel
to a discrete time channel. We take the usual approach of the sampling theorem. Assume
that the baseband input u(t) is bandlimited to W/2.

u(t) =       un sinc(W t − n)                     (25)
n

where un is deﬁned to be u(n/W ). Using (21), the baseband output is given by

v(t) =         un       βj (t) exp{−i2πfc τj (t)} sinc[tW − τj (t)W − n]   (26)
n        j

The sampled outputs at multiples of 1/W , i.e. vm = v(m/W ) are then given by

vm =        un        βj (m/W ) exp{−i2πfc τj (m/W )} sinc[m − n − τj (m/W )W ]       (27)
n         j

7
Let k = m − n. Then

vm =        um−k        βj (m/W ) exp{−i2πfc τj (m/W )} sinc[k − τj (m/W )W ]    (28)
k           j

By deﬁning

gk,m =       βj (m/W ) exp{−i2πfc τj (m/W )} sinc[k − τj (m/W )W ],       (29)
j

(28) can be written in the simple form

vm =       gk,m um−k                            (30)
k

We denote gk,m as the k th (complex) channel ﬁlter tap at time m. As we discuss later,
the number of channel ﬁlter taps (i.e., diﬀerent values of k) for which gk,m is signiﬁcantly
non-zero is usually quite small. If for each k, the k th tap is unchanging with m, then
the channel is linear time-invariant. If each tap changes slowly with m, then we call the
channel slowly time-varying. As seen in the next subsection, cellular systems and most of
the other wireless systems of current interest are slowly time-varying.
Due to Doppler shift, the bandwidth of the output v(t) is generally slightly larger than the
bandwidth W/2 of the input u(t), and thus the output samples {vm } do not fully represent
the output waveform. This problem is usually ignored in practice, where baseband ﬁlters
are not narrow enough to eliminate the Doppler shifted part of the received waveform
outside of W/2. Since the taps are slowly varying, they behave like an LTI ﬁlter over
the periods of interest and essentially model the channel. Also, it is very convenient for
the sampling rate of the input and output to be the same. Alternatively, it would be
possible to sample the output at twice the rate of the input. This would recapture all the
information in the received waveform but would essentially require doubling the number
of taps.

3    Time and Frequency Coherence; Multipath Spread
Recall from section 2 that signiﬁcant changes in βj occur over periods of seconds or more.
Signiﬁcant changes in the phase for each path occur at intervals of 1/(2D), where D
is the Doppler spread for the channel. Multipath fading occurs because diﬀerent paths
have diﬀerent Doppler shifts. Typical intervals for such changes are on the order of 10
msec. Finally, changes in the sinc term of (26) due to the time variation of each τj (t) are
proportional to the bandwidth, whereas those in the phase are proportional to the carrier
frequency, which is much larger. Thus, the fastest changes in the ﬁlter taps occur because
of the phase changes, and these are signiﬁcant over delay changes of 1/(2D).
The time coherence, Tc , of a narrowband wireless channel was deﬁned (in an order of
magnitude sense) as the interval over which gm,n changes signiﬁcantly. What we have

8
found, then, is the important relation:
1
Tc =                                          (31)
2D
This is a somewhat imprecise relation, since diﬀerent paths have diﬀerent Doppler shifts,
and the largest Doppler shifts may belong to paths that are too weak to make a diﬀerence.
Another important general parameter of a wireless system is the multipath spread, L,
deﬁned as the diﬀerence in propagation time between the longest and shortest path,
where we assume, in all of the above sums over diﬀerent paths, that only the signiﬁcant
paths are included. Thus,
L = max τj (t) − min τj (t)                              (32)
j               j

This is deﬁned as a function of t, but we regard it as an order of magnitude quantity,
like the time coherence and Doppler shift. If a cell or LAN has a linear extent of a few
kilometers or less, it is very unlikely to have path lengths that diﬀer by more than 300 to
600 meters. This corresponds to path delays of one or two µsec. As cells become smaller
due to increased cellular usage, L also shrinks.
The bandwidths of cellular systems range between several hundred kH and several MH,
and thus, for the above multipath spread values, all the path delays in (29) lie within the
peaks of 2 or 3 sinc functions; more often, they lie within a single peak. Adding a few
extra taps to each channel ﬁlter because of the slow decay of the sinc function, we see
that cellular channels can be represented with at most 4 or 5 channel ﬁlter taps.
When we study modulation and detection for these channels, we shall see that the re-
ceiver must estimate the values of these channel ﬁlter taps. These taps are estimated via
transmitted and received waveforms, and thus the receiver makes no explicit use of (and
usually does not have) any information about individual path delays and path strengths.
This is why we have not studied the details of propagation over multiple paths with com-
plicated types of reﬂection mechanisms. All we really need is the aggregate values of gross
There is one additional gross mechanism called frequency coherence. Wireless channels
change both in time and frequency. The time coherence shows us how quickly the channel
changes in time, and similarly, the frequency coherence shows how quickly it changes in
frequency. We ﬁrst understood about channels changing in time, and correspondingly
about the duration of fades, by studying the simple example of a direct path and a single
reﬂected path. That same example shows us how channels change with frequency.
ˆ
For a particular path, g (f, t) has linear phase in f . For multiple paths, there is a diﬀeren-
tial phase, 2πf (τj (t) − τk (t)). This diﬀerential phase causes selective fading in frequency.
This says that not only does Er (f, t) change signiﬁcantly when t changes by 1/(2D), but
also when f changes by 1/2L. This argument extends to an arbitrary number of paths,
so the frequency coherence, Fc is given by
1
Fc =                                          (33)
2L

9
This relationship, like (31) is intended as an order of magnitude relation, essentially point-
ing out that frequency coherence is reciprocal to multipath spread. When the bandwidth
of the input is considerably less than Fc , the channel is usually referred to as ﬂat fading,
and, in essence, a single channel ﬁlter tap is suﬃcient to represent the channel. Note that
ﬂat fading is not a property of the channel alone, but of the relationship between W and
Fc .

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