Lecture ¡1 Wireless ¡Channel I-Hsiang Wang ihwang@ntu.edu.tw 2/20, 2014
Wireless ¡channels ¡vary ¡at ¡two ¡scales Channel quality Time • Large-scale fading: path loss, shadowing, etc. • Small-scale fading: constructive/destructive interference 2
Large-‑Scale ¡Fading • Path loss and Shadowing ∝ 1 - In free space, received power r 2 1 - 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 3
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 λ - Typical carrier frequency ~ 1GHz ⇒ λ ≈ c/f = 0 . 3m = • Variation over time: order of hundreds of microseconds • Critical for design of communication systems 4
Plot • 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 5
Outline • Physical modeling of wireless channels • Deterministic Input-output model • Time and frequency coherence • Statistical models 6
Physical ¡Model: ¡ Warm-‑up ¡Examples
Physical ¡Model: ¡Simple ¡Example ¡1 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 8
Physical ¡Model: ¡Simple ¡Example ¡1 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 T d Phase Di ff erence between the two sinusoids: Delay Spread : ⇢ 2 π f (2 d − r ) � − 2 π fr = 2 π (2 d − r ) − r ∆ θ = + π f + π difference c c c between delays ( 2 n π , constructive interference = (2 n + 1) π , destructive interference 9
Delay ¡Spread ¡and ¡Coherence ¡Bandwidth • Delay spread � : difference between delays of paths T d • If frequency f change by �� 1 / (2 T d ) � � , then the combined received sinusoid move from peak to valley • Therefore, the frequency-variation scale is of the order of 1 T d W c := 1 • Coherence bandwidth T d 10
Physical ¡Model: ¡Simple ¡Example ¡2 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 11
Physical ¡Model: ¡Simple ¡Example ¡2 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 Time-varying amplitude of the original input waveform 12
Physical ¡Model: ¡Simple ¡Example ¡2 t ✓ vt ◆ c + r 0 − d 2 α r 0 + vt sin 2 π f Time-varying envelope c Doppler Spread D s = 2 fv c Difference of the Doppler shifts of r 0 /v Time-variation scale: the two paths, cause this variation (seconds or minutes), over time. much smaller than that of c/fv Time-variation scale: �� (ms) the second term 13
Doppler ¡Spread ¡and ¡Coherence ¡Time • Mobility causes time-varying delays (Doppler shift) • Doppler spread : difference between Doppler shifts of D s multiple signal paths • If time t change by � � 1 / (2 D s ) � � , then the combined received sinusoidal envelope move from peak to valley • Therefore, the time-variation scale is of the order of 1 D s T c := 1 • Coherence time D s 14
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 15
Physical ¡Model: ¡ Input/Output ¡Relations
Physical ¡Input/Output ¡Model • Wireless channels as linear time-varying systems: X y ( t ) = a i ( t ) x ( t − τ i ( t )) i a i ( t ): gain of path i τ i ( t ): delay of path i • Recall Example 2: d r ( t ) v x ( t ) = cos 2 π ft | α | τ 1 ( t ) = r 0 + vt a 1 ( t ) = r 0 + vt c | α | τ 2 ( t ) = 2 d − r 0 − vt π a 2 ( t ) = − 2 d − r 0 − vt 2 π f c 17
Physical ¡Input/Output ¡Model • Wireless channels as linear time-varying systems: X y ( t ) = a i ( t ) x ( t − τ i ( t )) i a i ( t ): gain of path i τ i ( t ): delay of path i • Impulse response: X h ( τ , t ) y ( t ) = a i ( t ) x ( t − τ i ( t )) x ( t ) i 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 18
Passband–Baseband ¡Conversion S ( f ) 1 f – f c – W W W f c – W – f c + f c + 2 2 2 2 • Communications takes place in a passband - Carrier frequency f c - Bandwidth W < 2 f c - Real signal s ( t ) 19
Passband–Baseband ¡Conversion S ( f ) 1 f – f c – W W W f c – W – f c + f c + 2 2 2 2 S b ( f ) � √ 2 S � f + f c � f + f c > 0 � S b � f � = √ 2 0 f + f c ≤ 0 � f W W – 2 2 20
Passband–Baseband ¡Conversion √ 2 cos 2 π f c t √ 2 cos 2 π f c t ℜ [ s b ( t )] ℜ [ s b ( t )] 1 X X – W W 2 2 + s ( t ) ℑ [ s b ( t )] ℑ [ s b ( t )] 1 X X – W W 2 2 – √ 2 sin 2 π f c t – √ 2 sin 2 π f c t √ s � t � = 1 s b � t � e j2 � f c t + s ∗ b � t � e − j2 � f c t � s b � t � e j2 � f c t � � � 2 ℜ √ = 2 21
Baseband ¡System ¡Architecture √ 2 cos 2 π f c t √ 2 cos 2 π f c t 1 ℜ [ x b ( t )] ℜ [ y b ( t )] X X – W W 2 2 x ( t ) y ( t ) + h ( τ , t ) 1 ℑ [ x b ( t )] ℑ [ y b ( t )] X X – W W 2 2 – √ 2 sin 2 π f c t – √ 2 sin 2 π f c t X a b y b ( t ) = i ( t ) x b ( t − τ i ( t )) , i i ( t ) := a i ( t ) e − j 2 π f c τ i ( t ) where a b 22
Continuous-‑time ¡Baseband ¡Model • Complex baseband equivalent channel: X a b y b ( t ) = i ( t ) x b ( t − τ i ( t )) x b ( t ) h b ( τ , t ) i X a b h b ( τ , t ) = i ( t ) δ ( τ − τ i ( t )) , i i ( t ) := a i ( t ) e − j 2 π f c τ i ( t ) where a b • Frequency response: shifted from passband to baseband H b ( f ; t ) = H ( f + f c ; t ) • Each path is associated with a delay and a complex gain 23
Modulation ¡and ¡Sampling • Modern communication systems are digitized, (partially) thanks to sampling theorem • Our baseband signal can be represented as follows: X x b ( t ) = x [ n ]sinc( Wt − n ) , n sinc( t ) := sin π t x [ n ] := x n ( n/W ) , π t 24
Modulation ¡and ¡Sampling √ 2 cos 2 π f c t √ 2 cos 2 π f c t 1 ℜ [ y b ( t )] ℜ [ y [ m ]] ℜ [ x b ( t )] ℜ [ x [ m ]] X sinc ( Wt – n ) X –W W 2 2 x ( t ) y ( t ) + h ( τ , t ) 1 ℑ [ y b ( t )] ℑ [ x b ( t )] ℑ [ y [ m ]] ℑ [ x [ m ]] X X sinc ( Wt – n ) –W W 2 2 – √ 2 sin 2 π f c t – √ 2 sin 2 π f c t X y [ m ] = h l [ m ] x [ m − l ] , l X a b where h l [ m ] := i ( m/W )sinc [ l − τ i ( m/W ) W ] i 25
Discrete-‑Time ¡Baseband ¡Model • Discrete-time channel model X y [ m ] = h l [ m ] x [ m − l ] x [ m ] h l [ m ] l X a b h l [ m ] := i ( m/W )sinc [ l − τ i ( m/W ) W ] i • Note: the l -th tap h l contains contributions mostly for the paths that have delays that lie inside the bin (roughly) l � 2 W , l 1 1 W + W − 2 W 1 • System resolves the multipaths up to delays of W 26
Multipath ¡Resolution 1 X W a b h l [ m ] := i ( m/W )sinc [ l − τ i ( m/W ) W ] i • sinc( t ) vanish quickly outside of i = 0 Main contribution l = 0 the interval [-0.5, 0.5] (roughly) i = 1 Main contribution l = 0 • The peak of the i -th translated sinc lies at τ i i = 2 Main contribution l = 1 • To contribute significantly to h l , i = 3 Main contribution l = 2 the delay must fall inside l � 2 W , l 1 1 W + i = 4 Main contribution l = 2 W − 2 W l 0 1 2 27
Time ¡and ¡Frequency ¡ Coherence
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