tsks04&digital&communication - linköping university · 2016-01-22 ·...

TSKS04 Digital CommunicationContinuation Course

Lecture 2

Complex Baseband Representation and Bandwidth

Emil Björnson

Department of Electrical Engineering (ISY)Division of Communication Systems

Complex Baseband Representation

Real-valued baseband information signals (bandwidth 𝐵/2):𝑠%(𝑡) and 𝑠)(𝑡)

Real-valued passband signal (𝑓+ ≫ 𝐵):𝑠 𝑡 = 𝑠% 𝑡 2 cos 2𝜋𝑓+𝑡 − 𝑠) 𝑡 2 sin 2𝜋𝑓+𝑡

2016-01-22 TSKS04 Digital Communication Continuation Course - Lecture 2 2

In-phase Quadrature-phase

Complex Baseband Representation or Complex envelope�̃� 𝑡 = 𝑠% 𝑡 + 𝑗𝑠)(𝑡)

Relationships:𝑠 𝑡 = 𝑅𝑒 2�̃� 𝑡 𝑒:;<=>?

Processing in Complex Baseband

§ Signal processing at baseband§ Use general purpose hardware§ Sampling frequency depend on baseband signal

§ Compact descriptions of algorithms and analysis


Create complex baseband signal

Modulate to carrier

frequencySend over channel

Demodulate to baseband

Sample complex baseband signal

Frequency Domain: Baseband and Passband

Fourier transforms: 𝑆A 𝑓 = ℱ �̃� 𝑡

𝑆 𝑓 =12𝑆A 𝑓 − 𝑓+ + 𝑆A∗ −𝑓 − 𝑓+


+𝑓+−𝑓+

0

𝑆A 𝑓

𝑆 𝑓

𝐵/2

𝐵

Inner Products of Complex Signals

Definition: Inner product of two complex signals 𝑎 𝑡 and 𝑏 𝑡 :

< 𝑎, 𝑏 > = L 𝑎 𝑡 𝑏∗ 𝑡 𝑑𝑡N

ON§ Property: < 𝑎 + 𝑏 , 𝑐 > = < 𝑎, 𝑐 > + < 𝑏, 𝑐 >

Parseval’s identity:

L 𝑎 𝑡 𝑏∗ 𝑡 𝑑𝑡N

ON= L 𝐴 𝑓 𝐵∗ 𝑓 𝑑𝑓

N

ON

Consequences:

< 𝑎,𝑏 > = L 𝐴 𝑓 𝐵∗ 𝑓 𝑑𝑓N

ON= < 𝐴,𝐵 >

𝑎 ; = < 𝑎, 𝑎 > = < 𝐴,𝐴 > = 𝐴 ;


Orthogonality between I- and Q-parts

Two passband signals:𝑎 𝑡 = 𝑎% 𝑡 2 cos 2𝜋𝑓+𝑡 − 𝑎) 𝑡 2 sin 2𝜋𝑓+𝑡𝑏 𝑡 = 𝑏% 𝑡 2 cos 2𝜋𝑓+𝑡 − 𝑏) 𝑡 2 sin 2𝜋𝑓+𝑡


Can treat I and Q parts separately< 𝑎,𝑏 > = < 𝑎%, 𝑏% > + < 𝑎), 𝑏) > = 𝑅𝑒 < 𝑎R, 𝑏S >

𝐸U = 𝑎 ; = 𝑎% ;+ 𝑎);= 𝑎R ; = 𝐸UR

Recall: Linearly Modulated Signal

Pulse-amplitude modulated signal:

𝑆 𝑡 = V V𝑆W 𝑛 𝜙W(𝑡 − 𝑛𝑇 −Ψ)\

W]+

N

^]ON


𝑛th data symbol(stochastic as before)𝑛 ∈ … , −1,0, +1, … = ℤ

𝑖th basis function(not necessarily time-limited)

Time delayRandom delay

How to select 𝑁 and 𝑇?

Recall: Sine-Shaped Basis Functions Same frequency (𝑁 = 2)


Essentially the same energy spectra!90%-bandwidth:

2/𝑇

Guard bands to not interfere with other

systems

Linear Modulation in Complex Baseband

Linear modulation with sine-shaped basis functions:

𝑆 𝑡 = V 𝑆+ 𝑛2𝑇 cos 2𝜋𝑓+𝑡 𝐼[^f,^fgf]

N

^]ON+ 𝑆; 𝑛

2𝑇 sin 2𝜋𝑓+𝑡 𝐼[^f,^fgf]

Compare with:

𝑠 𝑡 = 𝑠% 𝑡 2 cos 2𝜋𝑓+𝑡 − 𝑠) 𝑡 2 sin 2𝜋𝑓+𝑡

𝑠% 𝑡 = V 𝑆+ 𝑛1𝑇 𝐼[^f,^fgf]

N

^]ON, 𝑠) 𝑡 = − V 𝑆; 𝑛

1𝑇 𝐼[^f,^fgf]

N

^]ON


Pulse in PAM Pulse in PAM

Recall: Nyquist Criterion for ISI-Free CommunicationSampling of 𝑦 𝑡 = ∑ 𝑥 𝑘 𝑝(𝑡 − 𝑘𝑇)o at time 𝑛𝑇:

𝑧 𝑛 = 𝑦 𝑛𝑇 =V𝑥 𝑘 𝑝(𝑛𝑇 − 𝑘𝑇)o

Inter-symbol interference (ISI): 𝑧[𝑛] depends on 𝑥[𝑘] for 𝑘 ≠ 𝑛.

Goal: ISI-free communication!𝑧 𝑛 = 𝐾 ⋅ 𝑥 𝑛 for constant 𝐾 ≠ 0.


Achieved when 𝑝 𝑛𝑇 = 𝐾𝛿 𝑛 ⟺ 1𝑇

V 𝑃 𝑓 −𝑚𝑇 = 𝐾

N

x]ON

Nyquist ISI criterion: Constant!

A pulse with bandwidth 1/(2T) has to be a sinc to be Nyquist


A pulse 𝑝 𝑡 is said to be 𝐍𝐲𝐪𝐮𝐢𝐬𝐭 if V 𝑃 𝑓 −𝑚𝑇

N

x]ON

is constant

A pulse with bandwidth more than 1/(2T)can have many shapes to be Nyquist


A pulse 𝑝 𝑡 is said to be 𝐍𝐲𝐪𝐮𝐢𝐬𝐭 if V 𝑃 𝑓 −𝑚𝑇

N

x]ON

is constant

Sine-Shaped Basis Functions: A bad choice?

§ Pulse shape: 𝑝 𝑡 = +f 𝐼[�,f]

§ Fourier transform: 𝑃 𝑓 = − +f sinc 𝑓𝑇

§ Strictly speaking: Bandwidth 𝐵 = ∞§ Guard bands needed?

§ 90%-energy bandwidth: 1/𝑇§ 95%-energy bandwidth: 2/𝑇§ 99%-energy bandwidth: 10/𝑇


Better choices from TSKS01?• Sinc function• Raised cosine (𝛼 excess bandwidth)

Smaller if we increase 𝑇

Sine-Shaped Basis Functions (1/3) 𝑁 different frequencies

§ How closely can we place sine-shaped basis functions?

𝜙+ 𝑡 = 2/𝑇 cos 2𝜋𝑓+𝑡 , for 0 ≤ 𝑡 < 𝑇,

𝜙; 𝑡 = 2/𝑇 cos 2𝜋𝑓;𝑡 , for 0 ≤ 𝑡 < 𝑇,

𝜙� 𝑡 = 2/𝑇 sin 2𝜋𝑓;𝑡 , for 0 ≤ 𝑡 < 𝑇,

Orthogonal if:

L 𝜙+ 𝑡f

�𝜙; 𝑡 𝑑𝑡 = L cos 2𝜋(𝑓++𝑓;)𝑡 + cos 2𝜋(𝑓+−𝑓;)𝑡

f

�𝑑𝑡 = 0

L 𝜙+ 𝑡f

�𝜙� 𝑡 𝑑𝑡 = L sin 2𝜋(𝑓++𝑓;)𝑡 + sin 2𝜋(𝑓+−𝑓;)𝑡

f

�𝑑𝑡 = 0


2𝑓+𝑇, 2𝑓;𝑇 integers

𝑓+𝑇, 𝑓;𝑇 integers



Independent basis functions:Still substantial overlap in frequency!

Φ+ 𝑓 ;Φ; 𝑓 ; = Φ� 𝑓 ;

Guard band = 𝛽/𝑇

Distance between carriers:𝑓+ − 𝑓; = 𝑘/𝑇, 𝑘 = integer



Guard band = 𝛽/𝑇(𝑁+ 1)/𝑇

Guard band = 𝛽/𝑇

Total bandwidth:

𝐵 =𝑁 + 1 + 2𝛽

𝑇

Single-Carrier vs. Multi-Carrier Signals

§ Available passband bandwidth: 𝐵

§ Single-carrier transmission§ One carrier frequency 𝑓+§ Use raised cosine pulse with bandwidth 𝐵 = +g�

f§ Symbol time 𝑇 = 1 + 𝛼 /𝐵

§ Multi-carrier transmission§ Use 𝑁 carrier frequencies 𝑓+, 𝑓; ,… , 𝑓\§ Total bandwidth: 𝐵 = \g+g;�

f§ Symbol time 𝑇 = (𝑁 + 1 + 2𝛽)/𝐵 ≈ 𝑁/𝐵 if 𝑁 is large


Example: OFDM systems:

𝑁 ∝ 1000

Around 1/𝐵 complex symbols/s in both cases (e.g., QPSK, 16-QAM)

Impact of Channel Filter

Classification:§ Non-dispersive: 𝑔� 𝑡 = 𝛿(𝑡)

§ Time-dispersive: Spreads signal in time


𝑔� 𝑡𝑠(𝑡) 𝑟 𝑡 = (𝑠 ∗ 𝑔�)(𝑡)

Delay spread:Time interval where 𝑔� 𝑡 ≠ 0

Approximately non-dispersive: Delay spread small compared to symbol time

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tsks04&digital&communication - linköping university · 2016-01-22 ·...

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