Principle of Communications, Fall 2017 Lecture 06 Wireless Communication I-Hsiang Wang ihwang@ntu.edu.tw National Taiwan University 2017/12/27,28
Recap • Lecture 05 explored wideband communications over wires • Point-to-point communication: single Tx/Rx pair • Physical modeling: ‣ Noise modeled as additive white Gaussian noise ‣ Frequency selectivity modeled as convolution with LTI filter • End-to-end equivalent discrete-time complex baseband channel • Techniques developed: ‣ Optimal detection principles at receiver (Lecture 03) ‣ Error-correction coding to achieve reliable communication in the presence of noise (Lecture 04) ‣ Interference mitigation techniques to combat inter-symbol interference (Lecture 05) • Key feature: channel is quite static and stationary over time. 2
Wireless Communication • Wireless is a shared medium, inherently di ff erent from wireline ‣ More than one pairs of Tx/Rx can share the same wireless medium ‣ ⟹ can support more users , but also more interference ‣ Signals: broadcast at Tx, superimposed at Rx ‣ ⟹ more paths from Tx to Rx (variation over frequency) ‣ Mobility of Tx and Rx ‣ ⟹ channel variation over time ‣ Fading : the scale of variation over time and frequency matters • Key challenges: interference and fading • Look at point-to-point communication and focus on fading ‣ Where does fading come from? ‣ How to combat fading? 3
Outline • Modeling of wireless channels ‣ Physical modeling ‣ Time and frequency coherence ‣ Statistical modeling • Fading and diversity ‣ Impact of fading on signal detection ‣ Diversity techniques 4
Part I. Modeling Wireless Channels Physical Models; Equivalent Complex Baseband Discrete-Time Models; Stochastic Models 5
Multi-Path Physical Model Signals are transmitted using EM waves at a certain frequency f c Far-field assumption: speed of light Tx-Rx distance � λ c � c f c Approximate EM signals as rays under the far-field assumption. Each path corresponds to a ray. The input-output model of the wireless channel (neglect noise) X y ( t ) = a i ( t ) x ( t − τ i ( t )) i 6
X y ( t ) = a i ( t ) x ( t − τ i ( t )) i For path i : a i ( t ) : channel gain (attenuation) of path i τ i ( t ) : propagation delay of path i Simplest example: single line-of-sight (LOS) y ( t ) = α r x ( t − r c ) x ( t ) r τ ( t ) = r a ( t ) = α (free space) ; r c 7
X y ( t ) = a i ( t ) x ( t − τ i ( t )) i Example: single LOS with a reflecting wall d r Path 1: a 1 ( t ) = α r ; τ 1 ( t ) = r c Path 2: a 2 ( t ) = − τ 2 ( t ) = 2 d − r 2 d − r ; α c 8
X y ( t ) = a i ( t ) x ( t − τ i ( t )) i Example: single LOS with a reflecting wall and moving Rx d r ( t ) = r 0 + vt v τ 1 ( t ) = r 0 + vt Path 1: a 1 ( t ) = r 0 + vt ; α c τ 2 ( t ) = 2 d − r 0 − vt Path 2: a 2 ( t ) = − 2 d − r 0 − vt ; α c 9
Linear Time Varying Channel Model X y ( t ) = a i ( t ) x ( t − τ i ( t )) h ( τ ; t ) x ( t ) i Impulse response: X h ( τ ; t ) = a i ( t ) δ ( τ − τ i ( t )) i ˘ Frequency response: X a i ( t ) e − j 2 π f τ i ( t ) h ( f ; t ) = i Equivalent baseband model can be derived, similar to the derivation in wireline communication 10
Continuous-Time Baseband Model X a b y b ( t ) = i ( t ) x b ( t − τ i ( t )) x b ( t ) h b ( τ ; t ) i Impulse response: h b ( τ ; t ) = h ( τ ; t ) e − j2 π f c τ a b i ( t ) , a i ( t ) e − j2 π f c τ i ( t ) X a b = i ( t ) δ ( τ − τ i ( t )) i ˘ h b ( f ; t ) = ˘ Frequency response: h ( f + f c ; t ) The gain of each path is rotated with a phase 11
Discrete-Time Baseband Model X v m = h l [ m ] u m − l h l [ m ] u m l Z ∞ Impulse response: h ` [ m ] , h b ( ⌧ ; mT ) g ( ` T − ⌧ ) d ⌧ −∞ X a b = i ( mT ) g ( ` T − ⌧ i ( mT )) i Recall: g ( t ) is the pulse used in pulse shaping examples: sinc pulse, raised cosine pulse, etc. Observation: The ` -th tap h ` [ m ] majorly consists of the aggregation of paths with delay lying inside the “delay bin” ⌧ i ( mT ) ∈ ` T − T 2 , ` T + T ⇥ ⇤ 2 12
τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7 τ 8 delay 0 3 T T 2 T 13
ℓ = 0 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7 τ 8 delay 0 3 T T 2 T 14
ℓ = 1 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7 τ 8 delay 0 3 T T 2 T 15
ℓ = 2 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7 τ 8 delay 0 3 T T 2 T 16
ℓ = 3 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7 τ 8 delay 0 3 T T 2 T 17
Path resolution capability depends on the operating bandwidth τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 τ 7 τ 8 delay 0 3 T T 2 T 18
X v m = h ` [ m ] u m − ` h ` [ m ] u m ` X a b h ` [ m ] = i ( mT ) g ( ` T − ⌧ i ( mT )) i X a i ( mT ) e − j2 π f c τ i ( mT ) g ( ` T − ⌧ i ( mT )) = i X a i ( mT ) e − j2 ⇡ f c ⌧ i ( mT ) ≈ i ∈ ` ��� ����� ��� Difference in phases (over the paths that contribute significantly to the tap), causes variation of the tap gain 19
Large-scale Fading • Path loss and Shadowing ‣ In free space, received power ∝ r − 2 ‣ With reflections and obstacles, can attenuate faster than r − 2 • Variation over time: very slow, order of seconds • Critical for coverage and cell-cite planning 20
Multi-path (Small-scale) 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 λ c ‣ Typical carrier frequency ~ 1GHz ⇒ λ c ≈ c/f c = 0 . 3 � = • Variation over time: order of hundreds of microseconds • Critical for design of communication systems 21
Fading over Frequency d r Transmitted Waveform (electric field): cos 2 π ft t − r ⇣ ⌘ Received Waveform (path 1): α r cos 2 π f c ✓ ◆ t − 2 d − r α Received Waveform (path 2): − 2 d − r cos 2 π f c = ⇒ Received Waveform (aggregate): ✓ ◆ t − 2 d − r t − r ⇣ ⌘ α α r cos 2 π f 2 d − r cos 2 π f − c c 22
d r Transmitted Waveform (electric field): cos 2 π ft Received Waveform (aggregate): ✓ ◆ t − 2 d − r t − r ⇣ ⌘ α α r cos 2 π f 2 d − r cos 2 π f − c c Delay Spread T d Phase Di ff erence between the two sinusoids: delay differences ⇢ 2 π f (2 d − r ) � − 2 π fr = 2 π (2 d − r ) − r ∆ θ = + π f + π c c c ( 2 n π , constructive interference = (2 n + 1) π , destructive interference 23
Variation in Frequency Domain neglect dependency on time X y ( t ) = a i ( t ) x ( t − τ i ( t )) h ( τ ; t ) x ( t ) i Frequency response: ˘ X a i e − j2 πτ i f h ( f ) = i Frequency variation causes variation in phase shift. Phase di ff erence causes constructive or destructive interference. Phase di ff erence: � 2 π f Delay Spread i | τ i − τ ˜ i | 2 π f max i 6 =˜ T d � max i | τ i − τ ˜ i | i ̸ =˜ Frequency change by , channel changes drastically! 1 2 T d 24
������ ���� ��� ������ �������� ����� ������������������� ������ Coherence Bandwidth Coherence bandwidth: W c ∼ 1 T d From the perspective of the equivalent discrete-time model, for a system with operating (one-sided) bandwidth W : W c � 2 W = ) W c < 2 W = ) Note: this is a rough qualitative classification 25
Same channel, di ff erent operating bandwidth (b) (a) 0.004 10 0.003 0 0.002 –10 Power spectrum 0.001 (linear scale) Amplitude –20 0 (dB) –0.001 –30 –0.002 –40 –0.003 –50 –0.004 200 MHz –60 –0.005 –0.006 –70 0 50 100 150 200 250 300 350 400 450 500 550 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Time (ns) Frequency (GHz) (c) (d) 0.001 0 0.0008 –10 0.0006 Power specturm 0.0004 –20 (linear scale) Amplitude 0.0002 (dB) 0 –30 –0.0002 –40 –0.0004 –0.0006 40 MHz –50 –0.0008 –0.001 –60 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 50 100 150 200 250 300 350 400 450 500 550 Time (ns) Frequency (GHz) Larger bandwidth, more paths can be resolved 26
Fading over Time d r ( t ) = r 0 + vt v Transmitted Waveform (electric field): cos 2 π ft ✓ ◆ t − r ( t ) α Received Waveform (path 1): r ( t ) cos 2 π f c ✓ ◆ t − 2 d − r ( t ) α Received Waveform (path 2): − 2 d − r ( t ) cos 2 π f c = ⇒ Received Waveform (aggregate): ✓ ◆ ✓ ◆ t − r ( t ) t − 2 d − r ( t ) α α r ( t ) cos 2 π f 2 d − r ( t ) cos 2 π f − c c ⇣ � 1 − v t − r 0 1 + v t − 2 d − r 0 α h⇣ ⌘ i α ⌘ r 0 + vt cos 2 π f 2 d − r 0 − vt cos 2 π f = − c c c c 27
d v Approximation: distance to mobile Rx ⌧ distance to Tx = ⇒ Received Waveform (aggregate): ⇣ t − 2 d − r 0 � 1 − v t − r 0 1 + v h⇣ ⌘ i ⌘ α α = r 0 + vt cos 2 π f 2 d − r 0 − vt cos 2 π f − c c c c 2 α ✓ vt ◆ ✓ ◆ c + r 0 − d t − d r 0 + vt sin 2 π f sin 2 π f ≈ c c Time-invariant shift of the Time-varying amplitude original input waveform 28
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