Lecture1 WirelessChannel -...

Lecture 1Wireless Channel

I-Hsiang Wangihwang@ntu.edu.tw

2/20, 2014

Wireless channels vary at two scales

• Large-scale fading: path loss, shadowing, etc.

• Small-scale fading: constructive/destructive interference

11 2.1 Physical modeling for wireless channels

Figure 2.1 Channel qualityvaries over multipletime-scales. At a slow scale,channel varies due tolarge-scale fading effects. At afast scale, channel varies dueto multipath effects.

Channel quality

electromagnetic field impinging on the receiver antenna. This would have tobe done taking into account the obstructions caused by ground, buildings,vehicles, etc. in the vicinity of this electromagnetic wave.1

Cellular communication in the USA is limited by the Federal Commu-nication Commission (FCC), and by similar authorities in other countries,to one of three frequency bands, one around 0.9GHz, one around 1.9GHz,and one around 5.8GHz. The wavelength ! of electromagnetic radiation atany given frequency f is given by ! = c/f , where c = 3! 108 m/s is thespeed of light. The wavelength in these cellular bands is thus a fraction of ameter, so to calculate the electromagnetic field at a receiver, the locations ofthe receiver and the obstructions would have to be known within sub-meteraccuracies. The electromagnetic field equations are therefore too complex tosolve, especially on the fly for mobile users. Thus, we have to ask what wereally need to know about these channels, and what approximations might bereasonable.One of the important questions is where to choose to place the base-stations,

and what range of power levels are then necessary on the downlink and uplinkchannels. To some extent this question must be answered experimentally, butit certainly helps to have a sense of what types of phenomena to expect.Another major question is what types of modulation and detection techniqueslook promising. Here again, we need a sense of what types of phenomena toexpect. To address this, we will construct stochastic models of the channel,assuming that different channel behaviors appear with different probabilities,and change over time (with specific stochastic properties). We will return tothe question of why such stochastic models are appropriate, but for now wesimply want to explore the gross characteristics of these channels. Let us startby looking at several over-idealized examples.

1 By obstructions, we mean not only objects in the line-of-sight between transmitter andreceiver, but also objects in locations that cause non-negligible changes in the electro-magnetic field at the receiver; we shall see examples of such obstructions later.

Large-‐Scale Fading• Path loss and Shadowing- In free space, received power

- With reflections and obstacles, can attenuate faster than

• Variation over time: very slow, order of seconds

• Critical for coverage and cell-cite planning

Small-‐Scale Fading• Multipath fading: due to constructive and destructive

interference of the waves

• Channel varies when the mobile moves a distance of the order of the carrier wavelength - Typical carrier frequency ~ 1GHz

• Variation over time: order of hundreds of microseconds

• Critical for design of communication systems

=) � ⇡ c/f = 0.3m

Plan• Understand how physical parameters impact a wireless

channel from the communication system point of view. Physical parameters such as- Carrier frequency- Mobile speed- Bandwidth- Delay spread- etc.

• Start with deterministic physical models

• Progress towards statistical models

Outline• Physical modeling of wireless channels

• Deterministic Input-output model

• Time and frequency coherence

• Statistical models

Physical Model: Warm-‐up Examples

Physical Model: Simple Example 1

Transmitted Waveform (electric field): cos 2⇡ft

Received Waveform (path 1):

rcos 2⇡f

⇣t� r

Received Waveform (path 2): � ↵

2d� rcos 2⇡f

✓t� 2d� r

=) Received Waveform (aggregate):

rcos 2⇡f

⇣t� r

⌘� ↵

2d� rcos 2⇡f

✓t� 2d� r

Transmitted Waveform (electric field): cos 2⇡ft

Received Waveform (aggregate):

rcos 2⇡f

⇣t� r

⌘� ↵

2d� rcos 2⇡f

✓t� 2d� r

Phase Di↵erence between the two sinusoids:

�✓ =

⇢2⇡f(2d� r)

c+ ⇡

�� 2⇡fr

c= 2⇡

(2d� r)� r

cf + ⇡

(2n⇡, constructive interference

(2n+ 1)⇡, destructive interference

Delay Spread:difference

between delays

Delay Spread and Coherence Bandwidth• Delay spread! : difference between delays of paths

• If frequency f change by!! ! ! , then the combined received sinusoid move from peak to valley

• Therefore, the frequency-variation scale is of the order of

• Coherence bandwidth

1/(2Td)

Wc :=1

Received Waveform (path 1):

r(t)cos 2⇡f

✓t� r(t)

◆Transmitted Waveform (electric field): cos 2⇡ft

Received Waveform (path 2): � ↵

2d� r(t)cos 2⇡f

✓t� 2d� r(t)

r(t)cos 2⇡f

✓t� r(t)

◆� ↵

2d� r(t)cos 2⇡f

✓t� 2d� r(t)

r0 + vtcos 2⇡f

h⇣1� v

⌘t� r0

i� ↵

2d� r0 � vtcos 2⇡f

⇣1 +

⌘t� 2d� r0

r(t) = r0 + vt

Approximation: distance to mobile Rx ⌧ distance to Tx

Time-invariant shift of the original input waveform

Time-varying amplitude

r0 + vtcos 2⇡f

h⇣1� v

⌘t� r0

i� ↵

2d� r0 � vtcos 2⇡f

⇣1 +

⌘t� 2d� r0

⇡ 2↵

r0 + vtsin 2⇡f

r0 � d

◆sin 2⇡f

✓t� d

Difference of the Doppler shifts of the two paths, cause this variation over time. Time-variation scale:! ! (ms)

17 2.1 Physical modeling for wireless channels

Figure 2.5 The receivedwaveform oscillating atfrequency f with a slowlyvarying envelope at frequencyDs/2.

Er (t)

interference pattern and at its narrowest when the mobile is at a valley. Thus,the Doppler spread determines the rate of traversal across the interferencepattern and is inversely proportional to the coherence time of the channel.

We now see why we have partially ignored the denominator terms in (2.11)and (2.13). When the difference in the length between two paths changes bya quarter wavelength, the phase difference between the responses on the twopaths changes by !/2, which causes a very significant change in the overallreceived amplitude. Since the carrier wavelength is very small relative tothe path lengths, the time over which this phase effect causes a significantchange is far smaller than the time over which the denominator terms causea significant change. The effect of the phase changes is of the order ofmilliseconds, whereas the effect of changes in the denominator is of the orderof seconds or minutes. In terms of modulation and detection, the time-scalesof interest are in the range of milliseconds and less, and the denominators areeffectively constant over these periods.

The reader might notice that we are constantly making approximations intrying to understand wireless communication, much more so than for wiredcommunication. This is partly because wired channels are typically time-invariant over a very long time-scale, while wireless channels are typicallytime-varying, and appropriate models depend very much on the time-scales ofinterest. For wireless systems, the most important issue is what approximationsto make. Thus, it is important to understand these modeling issues thoroughly.

2.1.5 Reflection from a ground plane

Consider a transmit and a receive antenna, both above a plane surface suchas a road (Figure 2.6). When the horizontal distance r between the antennasbecomes very large relative to their vertical displacements from the ground

Time-varying envelope2↵

r0 + vtsin 2⇡f

r0 � d

Time-variation scale: (seconds or minutes), much smaller than that of the second term

Doppler Spread Ds =2fv

Doppler Spread and Coherence Time• Mobility causes time-varying delays (Doppler shift)

• Doppler spread : difference between Doppler shifts of multiple signal paths

• If time t change by!! ! ! , then the combined received sinusoidal envelope move from peak to valley

• Therefore, the time-variation scale is of the order of

• Coherence time

1/(2Ds)

Tc :=1

What we learned from the examples• Delay spread/coherence bandwidth and Doppler spread/

coherence time seem fundamental

• However, it is difficult to derive the explicit received waveform mathematically. - Out of scope – EM wave theory

• Instead, we construct useful input/output models, and take measurements to determine the parameters in the models

Physical Model: Input/Output Relations

Physical Input/Output Model• Wireless channels as linear time-varying systems:

• Recall Example 2:

y(t) =X

ai(t)x (t� ⌧i(t))

ai(t): gain of path i ⌧i(t): delay of path i

x(t) = cos 2⇡ft

a1(t) =|↵|

r0 + vt

⌧1(t) =r0 + vt

a2(t) =|↵|

2d� r0 � vt

⌧2(t) =2d� r0 � vt

� ⇡

Physical Input/Output Model• Wireless channels as linear time-varying systems:

• Impulse response:

• Frequency response:

y(t) =X

ai(t): gain of path i ⌧i(t): delay of path i

h (⌧, t)x(t) y(t) =X

h (⌧, t)x(t) y(t) =

h(⌧, t) =X

ai(t)� (⌧ � ⌧i(t))

H(f ; t) =X

ai(t)e�j2⇡f⌧i(t)

Passband–Baseband Conversion

• Communications takes place in a passband - Carrier frequency - Bandwidth - Real signal

23 2.2 Input/output model of the wireless channel

Figure 2.7 Illustration of therelationship between apassband spectrum S(f ) andits baseband equivalent Sb(f ).

Sb ( f )

S( f )

f –fc –

fc –W2

– fcW2

Since s!t" is real, its Fourier transform satisfies S!f "= S!!"f ", which meansthat sb!t" contains exactly the same information as s!t". The factor of

quite arbitrary but chosen to normalize the energies of sb!t" and s!t" to bethe same. Note that sb!t" is band-limited in #"W/2$W/2%. See Figure 2.7.To reconstruct s!t" from sb!t", we observe that

#2S!f "= Sb!f "fc"+S!

b!"f "fc"& (2.22)

Taking inverse Fourier transforms, we get

s!t"= 1#2

!sb!t"e

j2'fct + s!b!t"e"j2'fct

#2$!sb!t"e

j2'fct"& (2.23)

In terms of real signals, the relationship between s!t" and sb!t" isshown in Figure 2.8. The passband signal s!t" is obtained by modulating$#sb!t"% by

#2 cos2'fct and %#sb!t"% by "

#2 sin 2'fct and summing, to

get#2$!sb!t"ej2'fct

"(up-conversion). The baseband signal $#sb!t"% (respec-

tively %#sb!t"%) is obtained by modulating s!t" by#2 cos2'fct (respec-

tively "#2 sin 2'fct) followed by ideal low-pass filtering at the baseband

#"W/2$W/2% (down-conversion).Let us now go back to the multipath fading channel (2.14) with impulse

response given by (2.18). Let xb!t" and yb!t" be the complex basebandequivalents of the transmitted signal x!t" and the received signal y!t",respectively. Figure 2.9 shows the system diagram from xb!t" to yb!t". Thisimplementation of a passband communication system is known as quadratureamplitude modulation (QAM). The signal $#xb!t"% is sometimes called the

fcW < 2fcs(t)

Sb ( f )

S( f )

f –fc –

fc –W2

– fcW2

#2S!f "= Sb!f "fc"+S!

b!"f "fc"& (2.22)

s!t"= 1#2

!sb!t"e

#2$!sb!t"e

j2'fct"& (2.23)

get#2$!sb!t"ej2'fct

Sb ( f )

S( f )

f –fc –

fc –W2

– fcW2

#2S!f "= Sb!f "fc"+S!

b!"f "fc"& (2.22)

s!t"= 1#2

!sb!t"e

#2$!sb!t"e

j2'fct"& (2.23)

get#2$!sb!t"ej2'fct

22 The wireless channel

depend on time t, and we have the usual linear time-invariant channel withan impulse response

h!"#=!

ai$!"! "i#% (2.19)

For the time-varying impulse response h!"& t#, we can define a time-varyingfrequency response

H!f ' t# (=" "

!"h!"& t#e!j2)f" d" =

ai!t#e!j2)f"i!t#% (2.20)

In the special case when the channel is time-invariant, this reduces to theusual frequency response. One way of interpreting H!f ' t# is to think of thesystem as a slowly varying function of t with a frequency response H!f ' t#at each fixed time t. Corresponding, h!"& t# can be thought of as the impulseresponse of the system at a fixed time t. This is a legitimate and usefulway of thinking about many multipath fading channels, as the time-scaleat which the channel varies is typically much longer than the delay spread(i.e., the amount of memory) of the impulse response at a fixed time. In thereflecting wall example in Section 2.1.4, the time taken for the channel tochange significantly is of the order of milliseconds while the delay spread isof the order of microseconds. Fading channels which have this characteristicare sometimes called underspread channels.

2.2.2 Baseband equivalent model

In typical wireless applications, communication occurs in a passband*fc!W/2&fc+W/2+ of bandwidth W around a center frequency fc, thespectrum having been specified by regulatory authorities. However, mostof the processing, such as coding/decoding, modulation/demodulation,synchronization, etc., is actually done at the baseband. At the transmitter, thelast stage of the operation is to “up-convert” the signal to the carrier frequencyand transmit it via the antenna. Similarly, the first step at the receiver is to“down-convert” the RF (radio-frequency) signal to the baseband before furtherprocessing. Therefore from a communication system design point of view, itis most useful to have a baseband equivalent representation of the system.We first start with defining the baseband equivalent representation of signals.

Consider a real signal s!t# with Fourier transform S!f #, band-limited in*fc!W/2&fc+W/2+ with W< 2fc. Define its complex baseband equivalentsb!t# as the signal having Fourier transform:

Sb!f #=##

2S!f +fc# f +fc > 0&0 f +fc $ 0%

(2.21)

Figure 2.8 Illustration ofupconversion from sb(t) tos(t), followed bydownconversion from s(t)back to sb(t).

![sb(t)]

"[sb(t)]

![sb(t)]

"[sb(t)]

–!2 sin 2! fc t –!2 sin 2! fc

!2 cos 2! fc t!2 cos 2! fc

Figure 2.9 System diagramfrom the baseband transmittedsignal xb(t) to the basebandreceived signal yb(t). X

![xb(t)]

"[xb(t)]

![yb(t)]

"[yb(t)]

+x(t) y(t)

h(", t)

–!2 sin 2! fc t –!2 sin 2! fc

in-phase component I and "!xb"t#$ the quadrature component Q (rotatedby %/2). We now calculate the baseband equivalent channel. Substitutingx"t#=

#2!!xb"t#e j2%fct$ and y"t#=

#2!!yb"t#e j2%fct$ into (2.14) we get

!!yb"t#e j2%fct$ =!

ai"t#!!xb"t$ &i"t##ej2%fc"t$&i"t##$

= !"#!

ai"t#xb"t$ &i"t##e$j2%fc&i"t#

e j2%fct

' (2.24)

Similarly, one can obtain (Exercise 2.13)

"!yb"t#e j2%fct$= ""#!

e j2%fct

' (2.25)

Hence, the baseband equivalent channel is

yb"t#=!

abi "t#xb"t$ &i"t##( (2.26)

Sb ( f )

S( f )

f –fc –

fc –W2

– fcW2

#2S!f "= Sb!f "fc"+S!

b!"f "fc"& (2.22)

s!t"= 1#2

!sb!t"e

#2$!sb!t"e

j2'fct"& (2.23)

get#2$!sb!t"ej2'fct

Baseband System Architecture

Figure 2.8 Illustration ofupconversion from sb(t) tos(t), followed bydownconversion from s(t)back to sb(t).

![sb(t)]

"[sb(t)]

![sb(t)]

"[sb(t)]

–!2 sin 2! fc t –!2 sin 2! fc

Figure 2.9 System diagramfrom the baseband transmittedsignal xb(t) to the basebandreceived signal yb(t). X

![xb(t)]

"[xb(t)]

![yb(t)]

"[yb(t)]

+x(t) y(t)

h(", t)

–!2 sin 2! fc t –!2 sin 2! fc

in-phase component I and "!xb"t#$ the quadrature component Q (rotatedby %/2). We now calculate the baseband equivalent channel. Substitutingx"t#=

#2!!xb"t#e j2%fct$ and y"t#=

#2!!yb"t#e j2%fct$ into (2.14) we get

!!yb"t#e j2%fct$ =!

ai"t#!!xb"t$ &i"t##ej2%fc"t$&i"t##$

= !"#!

e j2%fct

' (2.24)

Similarly, one can obtain (Exercise 2.13)

"!yb"t#e j2%fct$= ""#!

e j2%fct

' (2.25)

Hence, the baseband equivalent channel is

yb"t#=!

abi "t#xb"t$ &i"t##( (2.26)

yb(t) =X

bi (t)xb (t� ⌧i(t)) ,

where a

bi (t) := ai(t)e

�j2⇡fc⌧i(t)

Continuous-‐time Baseband Model• Complex baseband equivalent channel:

• Frequency response: shifted from passband to baseband

• Each path is associated with a delay and a complex gain

xb(t) hb (⌧, t) yb(t) =X

bi (t)xb (t� ⌧i(t))

hb(⌧, t) =X

abi (t)� (⌧ � ⌧i(t)) ,

where abi (t) := ai(t)e�j2⇡fc⌧i(t)

Hb(f ; t) = H(f + fc; t)

Modulation and Sampling• Modern communication systems are digitized, (partially)

thanks to sampling theorem

• Our baseband signal can be represented as follows:

xb(t) =X

x[n]sinc(Wt� n),

x[n] := xn(n/W ), sinc(t) :=sin⇡t

Modulation and Sampling

XX![x[m]]

sinc (Wt – n)

"[x[m]]sinc (Wt – n)

h(!, t)

–W W

![xb(t)]

"[y[m]]

![y[m]]![yb(t)]

"[yb(t)]

y(t)x(t)

"[xb(t)]

–!2 sin 2" fc t –!2 sin 2" fc

!2 cos 2" fc t!2 cos 2" fc

The number of taps would be almost doubled because of the reduced sampleFigure 2.11 System diagramfrom the baseband transmittedsymbol x[m] to the basebandsampled received signal y[m].

interval, but it would typically be somewhat less than doubled since therepresentation would not spread the path delays so much.

Discussion 2.1 Degrees of freedom

The symbol x!m" is the mth sample of the transmitted signal; there areW samples per second. Each symbol is a complex number; we say that itrepresents one (complex) dimension or degree of freedom. The continuous-time signal x#t$ of duration one second corresponds toW discrete symbols;thus we could say that the band-limited, continuous-time signal has Wdegrees of freedom, per second.

The mathematical justification for this interpretation comes from thefollowing important result in communication theory: the signal space ofcomplex continuous-time signals of duration T which have most of theirenergy within the frequency band !#W/2%W/2" has dimension approx-imately WT . (A precise statement of this result is in standard com-munication theory text/books; see Section 5.3 of [148] for example.)This result reinforces our interpretation that a continuous-time signalwith bandwidth W can be represented by W complex dimensions persecond.

The received signal y#t$ is also band-limited to approximately W (dueto the Doppler spread, the bandwidth is slightly larger than W ) and has Wcomplex dimensions per second. From the point of view of communicationover the channel, the received signal space is what matters because itdictates the number of different signals which can be reliably distinguishedat the receiver. Thus, we define the degrees of freedom of the channelto be the dimension of the received signal space, and whenever we referto the signal space, we implicitly mean the received signal space unlessstated otherwise.

y[m] =X

hl[m]x[m� l],

where hl[m] :=X

bi (m/W )sinc [l � ⌧i(m/W )W ]

Discrete-‐Time Baseband Model• Discrete-time channel model

• Note: the l-th tap hl contains contributions mostly for the paths that have delays that lie inside the bin (roughly)

• System resolves the multipaths up to delays of

y[m] =X

hl[m]x[m� l]hl[m]x[m]

hl[m] :=X

abi (m/W )sinc [l � ⌧i(m/W )W ]

W� 1

Multipath Resolution

• sinc(t) vanish quickly outside of the interval [-0.5, 0.5] (roughly)

• The peak of the i-th translated sinc lies at

• To contribute significantly to hl, the delay must fall inside

Figure 2.10 Due to the decayof the sinc function, the i thpath contributes mostsignificantly to the !th tap ifits delay falls in the window"!/W ! 1/#2W $% !/W +1/#2W $&.

Main contribution l = 0

0 1 2l

at the output of the low-pass filter. Figure 2.11 shows the complete system.In practice, other transmit pulses, such as the raised cosine pulse, are oftenused in place of the sinc pulse, which has rather poor time-decay propertyand tends to be more susceptible to timing errors. This necessitates samplingat the Nyquist sampling rate, but does not alter the essential nature of themodel. Hence we will confine to Nyquist sampling.

Due to the Doppler spread, the bandwidth of the output yb!t" is generallyslightly larger than the bandwidth W/2 of the input xb!t", and thus the outputsamples #y$m%& do not fully represent the output waveform. This problem isusually ignored in practice, since the Doppler spread is small (of the orderof tens to hundreds of Hz) compared to the bandwidth W . Also, it is veryconvenient 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 ofthe input. This would recapture all the information in the received waveform.

hl[m] :=X

W� 1

Time and Frequency Coherence

Varying Channel Tap• The discrete-time baseband channel model is the

equivalent one in designing communication systems• It only matters how the taps hl[m] vary over time m and

carrier frequency fc

• l-th tap of the discrete-time baseband channel model

hl[m] :=X

ai (tm) e�j2⇡fc⌧i(tm)sinc [l � ⌧i(tm)W ] tm := mW

i2l-th delay bin

ai (tm) e�j2⇡fc⌧i(tm)

Difference in phases (over the paths that contribute significantly to the tap), causes variation of the tap gain

Frequency Variation

• Delay Spread

• Coherence Bandwidth

• For a system with bandwidth W

hl[m] ⇡X

i2l-th delay bin

Td := max

i,j|⌧i(t)� ⌧j(t)|

Wc :=1

Wc � W =) single tap, flat fading

Wc < W =) multiple taps, frequency-selective fading

Flat and Frequency-‐Selective Fading• Effective channel depends on both physical environment

(Wc) and operation bandwidth (W)

33 2.3 Time and frequency coherence

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76

–0.001–0.0008–0.0006–0.0004–0.0002

0 0.0002 0.0004 0.0006 0.0008

0 50 100 150 200 250 300 350 400 450 500 550

–70–0.006–0.005–0.004–0.003–0.002–0.001

0 0.001 0.002 0.003 0.004

50 100 150 200 250 300 350 400 450 500 5500 0.5

Time (ns)

40 MHz

Frequency (GHz)

200 MHz

phase causes selective fading in frequency. This says that Er!f" t# changesFigure 2.13 (a) A channel over200MHz is frequency-selective,and the impulse response hasmany taps. (b) The spectralcontent of the same channel.(c) The same channel over40MHz is flatter, and has forfewer taps. (d) The spectralcontents of the same channel,limited to 40MHz bandwidth.At larger bandwidths, the samephysical paths are resolved intoa finer resolution.

significantly, not only when t changes by 1/!4Ds#, but also when f changesby 1/!2Td#. This argument extends to an arbitrary number of paths, so thecoherence bandwidth, Wc, is given by

Wc =12Td

$ (2.47)

This relationship, like (2.44), is intended as an order of magnitude relation,essentially pointing out that the coherence bandwidth is reciprocal to themultipath spread. When the bandwidth of the input is considerably less thanWc, the channel is usually referred to as flat fading. In this case, the delayspread Td is much less than the symbol time 1/W , and a single channelfilter tap is sufficient to represent the channel. When the bandwidth is muchlarger than Wc, the channel is said to be frequency-selective, and it has tobe represented by multiple taps. Note that flat or frequency-selective fadingis not a property of the channel alone, but of the relationship between thebandwidth W and the coherence bandwidth Td (Figure 2.13).The physical parameters and the time-scale of change of key parameters of

the discrete-time baseband channel model are summarized in Table 2.1. Thedifferent types of channels are summarized in Table 2.2.

Larger bandwidth, more paths can be resolved

Time Variation

• Doppler Spread

• Coherence Time

• For a system with delay requirement (application dependent) T

hl[m] ⇡X

i2l-th delay bin

Ds := max

i,jfc|⌧i 0(t)� ⌧j

Tc :=1

Tc � T =) slow fading

Tc < T =) fast fading

Representitive Numbers

Table 2.1 A summary of the physical parameters of the channel and thetime-scale of change of the key parameters in its discrete-time basebandmodel.

Key channel parameters and time-scales Symbol Representative values

Carrier frequency fc 1GHzCommunication bandwidth W 1MHzDistance between transmitter and receiver d 1 kmVelocity of mobile v 64 km/hDoppler shift for a path D = fcv/c 50HzDoppler spread of paths corresponding to

a tap Ds 100HzTime-scale for change of path amplitude d/v 1 minuteTime-scale for change of path phase 1/!4D" 5msTime-scale for a path to move over a tap c/!vW " 20 sCoherence time Tc = 1/!4Ds" 2.5msDelay spread Td 1#sCoherence bandwidth Wc = 1/!2Td" 500 kHz

Table 2.2 A summary of the types of wirelesschannels and their defining characteristics.

Types of channel Defining characteristic

Fast fading Tc ! delay requirementSlow fading Tc " delay requirementFlat fading W !WcFrequency-selective fading W "WcUnderspread Td ! Tc

2.4 Statistical channel models

2.4.1 Modeling philosophy

We defined Doppler spread and multipath spread in the previous section asquantities associated with a given receiver at a given location, velocity, andtime. However, we are interested in a characterization that is valid over somerange of conditions. That is, we recognize that the channel filter taps {h$%m&}must be measured, but we want a statistical characterization of how manytaps are necessary, how quickly they change and how much they vary.Such a characterization requires a probabilistic model of the channel tap

values, perhaps gathered by statistical measurements of the channel. We arefamiliar with describing additive noise by such a probabilistic model (asa Gaussian random variable). We are also familiar with evaluating errorprobability while communicating over a channel using such models. These

Types of Channels

• Typical channels are underspread• Coherence time Tc depends on carrier frequency and

mobile speed, of the order of ms or more• Delay spread Td depends on distance to scatters and cell

size, of the order of ns (indoor) to µs (outdoor)

Table 2.1 A summary of the physical parameters of the channel and thetime-scale of change of the key parameters in its discrete-time basebandmodel.

Key channel parameters and time-scales Symbol Representative values

Carrier frequency fc 1GHzCommunication bandwidth W 1MHzDistance between transmitter and receiver d 1 kmVelocity of mobile v 64 km/hDoppler shift for a path D = fcv/c 50HzDoppler spread of paths corresponding to

a tap Ds 100HzTime-scale for change of path amplitude d/v 1 minuteTime-scale for change of path phase 1/!4D" 5msTime-scale for a path to move over a tap c/!vW " 20 sCoherence time Tc = 1/!4Ds" 2.5msDelay spread Td 1#sCoherence bandwidth Wc = 1/!2Td" 500 kHz

Table 2.2 A summary of the types of wirelesschannels and their defining characteristics.

Types of channel Defining characteristic

Fast fading Tc ! delay requirementSlow fading Tc " delay requirementFlat fading W !WcFrequency-selective fading W "WcUnderspread Td ! Tc

2.4 Statistical channel models

2.4.1 Modeling philosophy

We defined Doppler spread and multipath spread in the previous section asquantities associated with a given receiver at a given location, velocity, andtime. However, we are interested in a characterization that is valid over somerange of conditions. That is, we recognize that the channel filter taps {h$%m&}must be measured, but we want a statistical characterization of how manytaps are necessary, how quickly they change and how much they vary.Such a characterization requires a probabilistic model of the channel tap

values, perhaps gathered by statistical measurements of the channel. We arefamiliar with describing additive noise by such a probabilistic model (asa Gaussian random variable). We are also familiar with evaluating errorprobability while communicating over a channel using such models. These

Stochastic Models

• Continuous-time Passband ⟶ Discrete-time Baseband:

• Continuous-time Passband (Real)

• Continuous-time Baseband (Complex)

• Discrete-time Baseband (Complex)

Recap: Deterministic Modeling

h (⌧, t)x(t) y(t)

< {xb(t)}

= {xb(t)}

p2 cos 2⇡fct

�p2 sin 2⇡fct

p2 cos 2⇡fct

�p2 sin 2⇡fct

Filter⇥�W

2 , W2

Filter⇥�W

2 , W2

⇤= {yb(t)}

< {yb(t)}sinc(Wt� n)

sinc(Wt� n)

< {x[m]}

= {x[m]}

< {y[m]}

= {y[m]}

y(t) =X

ai(t)x (t� ⌧i(t)) ai(t): gain of path i ⌧i(t): delay of path i

yb(t) =X

bi (t)xb (t� ⌧i(t)) , a

bi (t) := ai(t)e

�j2⇡fc⌧i(t)

y[m] =X

hl[m]x[m� l], hl[m] :=X

Up-conversion Down-conversionModulation Sampling

• Thermal noise at the receiver:

• Continuous-time Passband (Real)

• Continuous-time Baseband (Complex)

• Discrete-time Baseband (Complex)

h (⌧, t)x(t) y(t)

< {xb(t)}

= {xb(t)}

p2 cos 2⇡fct

�p2 sin 2⇡fct

p2 cos 2⇡fct

�p2 sin 2⇡fct

Filter⇥�W

2 , W2

Filter⇥�W

2 , W2

⇤= {yb(t)}

< {yb(t)}sinc(Wt� n)

sinc(Wt� n)

< {x[m]}

= {x[m]}

< {y[m]}

= {y[m]}

ai(t): gain of path i ⌧i(t): delay of path iy(t) =

ai(t)x (t� ⌧i(t)) + w(t)

yb(t) =X

bi (t)xb (t� ⌧i(t)) + wb(t), a

bi (t) := ai(t)e

�j2⇡fc⌧i(t)

y[m] =X

hl[m]x[m� l] + w[m], hl[m] :=X

Additive White Noise Model

• Additive White Gaussian Noise (AWGN)- Standard modeling for thermal noise- {w(t)}: zero-mean (real) white Gaussian process with spectral

density N0/2- In other words,

• Discrete-time baseband equivalent noise:

E [w(0)w(t)] =N0

2�(t)

p2 cos 2⇡fct

�p2 sin 2⇡fct

Filter⇥�W

2 , W2

Filter⇥�W

2 , W2

< {w[m]}

= {w[m]}

• System is linear ⟶ can separate the noise out

• Rectangle filter in frequency ⟺ sinc in time

Equivalent Discrete-‐Time Baseband Noise

p2 cos 2⇡fct

�p2 sin 2⇡fct

Filter⇥�W

2 , W2

Filter⇥�W

2 , W2

< {w[m]}

= {w[m]}

◆rect

◆F�1

��! W sinc (Wt)

< {w[m]} =

�1w(t)

hp2W cos (2⇡fct) sinc (Wt�m)

| {z } m,1(t)

= {w[m]} =

�1w(t)

h�p2W sin (2⇡fct) sinc (Wt�m)

| {z } m,2(t)

= hw(t), m,1(t)i

= hw(t), m,2(t)i

Equivalent Discrete-‐Time Baseband Noise

< {w[m]} =

�1w(t)

hp2W cos (2⇡fct) sinc (Wt�m)

| {z } m,1(t)

= {w[m]} =

�1w(t)

h�p2W sin (2⇡fct) sinc (Wt�m)

| {z } m,2(t)

= hw(t), m,1(t)i

= hw(t), m,2(t)i

! ! ! ! ! ! ! ! forms an orthogonal set of waveforms.Fact.

{ m,1(t), m,2(t) | m 2 Z}

• The real and the imaginary parts of w[m] are - Both Gaussian with zero-mean and variance WN0/2- Independent and identically distributed (i.i.d.) over time (m)- “White” discrete-time processes

Circular Symmetric Complex Gaussian

! ! ! ! ! is a zero-mean white circular symmetric complex Gaussian process with auto-correlation function

Fact.{w[m] | m 2 Z}

R[m] := E [w[n+m]w[n]⇤] = WN0�[m]

w[m] = < {w[m]}+ j= {w[m]}

< {w[m]} ⇠ N✓0,

◆, = {w[m]} ⇠ N

(< {w[m]} ,= {w[m]}) : independent

X is circular symmetric if ei✓Xd= X for all ✓

() w[m] ⇠ CN (0,WN0) : circular symmetric complex Gaussian

Discrete Baseband Model with Noise

h (⌧, t)x(t) y(t)

< {xb(t)}sinc(Wt� n)

sinc(Wt� n)= {xb(t)}

< {x[m]}

= {x[m]}

p2 cos 2⇡fct

�p2 sin 2⇡fct

p2 cos 2⇡fct

�p2 sin 2⇡fct

Filter⇥�W

2 , W2

Filter⇥�W

2 , W2

⇤= {yb(t)}

< {yb(t)}

< {y[m]}

= {y[m]}

⌘hl[m]x[m] y[m] =

hl[m]x[m� l] + w[m]

w[m] ⇠ CN (0,WN0) ,

is the spectral density of white Gaussian process {w(t)}

Please refer to Appendix A for a review on complex

Gaussian random variables.

Fading

• Additive noise w[m]- Essentially completely random, no correlation over time- Largely depends on nature- Can be dealt with using digital (wired) communication techniques

• Filter taps hl[m]- Varying over time and frequency- Largely depends on nature- Why not use stochastic models for taps as well?

y[m] =X

hl[m]x[m� l] + w[m],

where hl[m] :=X

w[m] ⇠ CN (0,WN0)

Rayleigh Fading• Many small scattered paths for each tap:- Phase for each path is uniformly distributed over [0, 2π]

- For each path it is a circular symmetric random variable

• Each tap: sum of many small independent circular symmetric random variables- By Central Limit Theorem (CLT), we can model

- Zero-mean because of rich scattering

hl[m] ⇡X

hl[m] ⇠ CN�0,�2

Rician Fading• If there is a strong line-of-sight path, then model it as

• K-Factor! :- The larger it is, the more deterministic the channel will be.

hl[m] =

+ 1�le

j✓ +

�0,�2

Line-of-sight Scattered Multipath

Lecture1 WirelessChannel -...

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