Digital Band-pass Modulation

Digital Band-pass Modulation

PROF. MICHAEL TSAI

2011/11/10

Band-pass Signal Representation

• General form:

𝒈 𝒕 = 𝒂 𝒕 𝒄𝒐𝒔 𝟐𝝅𝒇

𝒕 + 𝝓 𝒕

• Envelope is always non-negative, or we can switch the phase by 180 degree

• This is called the canonical representation of a band- pass signal

Envelope Phase

𝑎 𝑡

2𝜋𝑓_𝑐𝑡 + 𝜙 𝑡 𝑔 𝑡

Band-pass Signal Representation

• 𝒈 𝒕 = 𝒂 𝒕 𝒄𝒐𝒔 𝟐𝝅𝒇

𝒕 + 𝝓 𝒕 can be re-arranged into

• 𝒈 𝒕 = 𝒈

𝒕 𝒄𝒐𝒔 𝟐𝝅𝒇

𝒕 − 𝒈

𝒕 𝒔𝒊𝒏 𝟐𝝅𝒇

𝒕

• 𝒈

𝒕 = 𝒂 𝒕 𝒄𝒐𝒔 𝝓 𝒕 and 𝒈

𝒕 = 𝒂 𝒕 𝒔𝒊𝒏 𝝓 𝒕

• 𝒈

𝒕 and 𝒈

𝒕 are called inphase and quadrature components of the signal g(t), respectively

• Then 𝒂 𝒕 = 𝒈

𝒕 + 𝒈

𝒕 and 𝝓 𝒕 = 𝒕𝒂𝒏

𝑰 𝒕

Band-pass Signal Representation

• We can also represent g(t) as

𝒈 𝒕 = 𝑹𝒆 𝒈 𝒕 𝒆𝒙𝒑 𝒋𝟐𝝅𝒇

𝒕

• 𝒈 𝒕 = 𝒈

𝒕 + 𝒋𝒈

𝒕

• 𝒈 𝒕 is called the complex envelope of the band-pass signal.

• This is to remove the annoying 𝒆𝒙𝒑 𝒋𝟐𝝅𝒇

𝒕 in the analysis.

𝑎 𝑡

𝜙 𝑡

𝑔 𝑡

Sinusoidal Functions’

Fourier Transform

• Complex exponential function

• 𝐹 exp 𝑗2𝜋𝑓

𝑡 = 𝛿(𝑓 − 𝑓

).

• Sinusoidal functions:

• cos 2𝜋𝑓

𝑡 =

2

exp 𝑗2𝜋𝑓

𝑡 + exp −𝑗2𝜋𝑓

𝑡

• 𝐹 cos 2𝜋𝑓

𝑡 =

2

𝛿 𝑓 − 𝑓

+ 𝛿 𝑓 + 𝑓

• sin 2𝜋𝑓

𝑡 =

2

exp 𝑗2𝜋𝑓

𝑡 − exp −𝑗2𝜋𝑓

𝑡

• 𝐹 sin 2𝜋𝑓

𝑡 =

𝛿 𝑓 − 𝑓

− 𝛿 𝑓 + 𝑓

5

f 𝐺(𝑓)

𝑓_𝑐

f 𝐺(𝑓)

𝑓_𝑐

−𝑓_𝑐 −𝑓_𝑐

f 𝐺(𝑓)

𝑓_𝑐

Band-pass Signal Transmitter

Signal Encoder

90 degree

shift

×

×

∼

Message

Source

Σ

cos⁡(2𝜋𝑓_𝑐𝑡)

sin⁡(2𝜋𝑓_𝑐𝑡) 𝑔_𝐼(𝑡)

𝑔_𝑄(𝑡)

𝑔 𝑡 = 𝑔_𝐼 𝑡 cos 2𝜋𝑓_𝑐𝑡 + 𝑔_𝑄 𝑡 sin 2𝜋𝑓_𝑐𝑡

+

+ Maps each bit into

𝑔_𝐼 𝑡 and 𝑔_𝑄 𝑡

Assumption

• The channel is linear: flat-fading channel.

• 𝐵

> 𝐵

• Negligible distortion to 𝑔(𝑡)

• The received signal s(t) is perturbed by AWGN

• noise w(t) ~𝑁 0,

•

2

is the PSD of the noise and also its variance (since it’s white)

AWGN Channel

Channel 𝐴

Band-pass

Signal g(t)

Σ

+

White Gaussian Noise 𝑤 𝑡

Received Signal plus Noise 𝑠 𝑡 + 𝑤(𝑡)

Channel path loss

or attenuation

𝑥 𝑡 = 𝑠 𝑡 + 𝑤 𝑡 = 𝐴_𝑐𝑔 𝑡 + 𝑤(𝑡)

Band-pass Signal Receiver

Band-pass Filter

90 degree

shift

×

×

∼

Message Sink Received

Signal plus Noise

cos⁡(2𝜋𝑓_𝑐𝑡)

sin⁡(2𝜋𝑓_𝑐𝑡)

𝑔_𝑄(𝑡) Filters out out-

of-band signals and noises

𝑥 𝑡 = 𝑠 𝑡 + 𝑛(𝑡)

Low-pass Filter

Signal Detector

Low-pass Filter

1

2 𝐴_𝑐𝑔_𝐼 𝑡 + 𝑛_𝐼 𝑡

1

2 𝐴_𝑐𝑔_𝑄 𝑡 + 𝑛_𝑄 𝑡 Mixer

Band-pass Filter

• The band-pass filter at the frontend filters out out-of-band signals and noises

1. Signal s(t) is within the band  not affected

2. White noise w(t) becomes narrowband noise n(t)

• Much smaller since now we only include noises within the band

• Still “white over the bandwidth of the signal”

3. Other signal (out-of-band) is filtered out

Band-pass Filter

𝑓_𝑐 𝑓 𝑓_𝑐 − 𝐵

2 𝑓_𝑐 + 𝐵

2 Other signals

Noise s(t)

Up-conversion (TX)

×

∼

𝑓 𝑓

𝐴 𝑡 exp 𝑗𝜃 𝑡 × A_ccos 2𝜋𝑓_𝑐𝑡 In time domain

In frequency domain

∗

Convolution

−𝑓_𝑐 𝑓_𝑐 𝑓

−𝑓_𝑐 𝑓_𝑐

Down-conversion (RX)

×

∼

s t A^′_ccos⁡(2𝜋𝑓_𝑐𝑡 + 𝜙) × A_ccos 2𝜋𝑓_𝑐𝑡 In time domain

In frequency domain

∗

Convolution

Low-pass Filter

−𝑓_𝑐 𝑓_𝑐 𝑓 𝑓

−𝑓_𝑐 𝑓_𝑐 𝑓

−2𝑓_𝑐 2𝑓_𝑐

Low-pass Filter

Signal Detector

• The signal detector:

• Observes complex representation of the received signal, 𝒈

𝒕 + 𝒏

𝒕 + 𝒋[𝒈

𝒕 + 𝒏

𝒕 ],

• For a duration of T seconds (symbol/bit period)

• And the make its best estimate of the corresponding transmitted signal 𝒈

𝒕 + 𝒋𝒈

𝒕

• 𝒈

𝒕 + 𝒋𝒈

𝒕  bit stream

Signal Detector

Time synchronization

• To simplify, we assume we have time synchronization between the TX and the RX

• Symbol boundary needs to be same for TX and RX

• In practice, a timing recovery circuit is required

𝑡 Where does each symbol start and end?

Coherent & non-coherent

• Sometimes, the receiver is phase-locked to the transmitter

• That means, the in TX and in RX generate 𝒄𝒐𝒔(𝟐𝝅𝒇

𝒕) with no phase difference.

• RX looks at the received signal to lock onto TX’s carrier

• When that happens, we say

• The receiver is a coherent receiver, carrying out coherent detection

• Otherwise, we say

• The receiver is a non-coherent receiver, carrying out non-coherent detection

∼

Basic forms of digital modulation

Amplitude Shift Keying

Frequency Shift Keying

Phase Shift Keying

Keying == Switching

(Binary) Amplitude Shift Keying (BASK)

• Fixed Amplitude/fixed frequency for a duration of 𝑻

to represent “1”

• No transmission to represent “0”

• Or, more formally,

• 𝑠

𝑡 = 𝐴

cos(2𝜋𝑓

𝑡)

• 𝑠

𝑡 = 0

(Binary) Phase Shift Keying (BPSK)

• Same amplitude, same frequency

• Send the original carrier to represent “1”

• Send an inverted carrier (phase difference 180 degrees) to represent “0”

• Or, more formally,

• 𝑠₁ 𝑡 = 𝐴_𝑐 cos(2𝜋𝑓_𝑐𝑡)

• 𝑠₀ 𝑡 = 𝐴_𝑐 cos 2𝜋𝑓_𝑐𝑡 + 𝜋 = −𝐴_𝑐 cos(2𝜋𝑓₀𝑡)

(Binary) Phase Shift

Keying (BPSK)

(Binary) Frequency Shift Keying (BFSK)

𝑇_𝑏

• Same amplitude

• Send a carrier at 𝒇

to represent “1”

• Send a carrier at 𝒇

to represent “0”

• Or, more formally,

• 𝑠₁ 𝑡 = 𝐴_𝑐 cos(2𝜋𝑓₁𝑡)

• 𝑠₀ 𝑡 = 𝐴_𝑐cos(2𝜋𝑓₀𝑡) for a duration of 𝑇_𝑏

(Binary) Frequency Shift Keying (BFSK)

• Usually we have 𝒇

= 𝒇

+ 𝚫𝐟, 𝐟

= 𝐟

− 𝚫𝐟

• 𝑠

𝑡 = 𝐴

cos[2𝜋 𝑓

+ Δ𝑓 𝑡]

• 𝑠

𝑡 = 𝐴

cos[2𝜋 𝑓

− Δ𝑓 𝑡]

• Then,

• 𝑠

𝑡 = 𝑅𝑒 𝐴

exp 𝑗2𝜋 𝑓

+ Δf 𝑡 = 𝑔 𝑡 exp 𝑗2𝜋𝑓

𝑡

• 𝑠

𝑡 = 𝑅𝑒 𝐴

exp 𝑗2𝜋 𝑓

− Δf 𝑡 = 𝑔 𝑡 exp 𝑗2𝜋𝑓

𝑡

• So,

• For “1”, 𝑔 𝑡 = 𝑔

𝑡 + 𝑗𝑔

𝑡 = 𝐴

exp⁡[−𝑗2𝜋Δ𝑓𝑡]

• For “0”,𝑔 𝑡 = 𝑔

𝑡 + 𝑗𝑔

𝑡 = 𝐴

exp⁡[+𝑗2𝜋Δ𝑓𝑡]

I Q

Coherent Detection of FSK and PSK signals

• Since 𝒇

is large compared to

𝒃

(symbol rate, or bit rate), we can say that the same signal energy 𝑬

is transmitted in a bit interval 𝑻

:

𝐸

= 𝑠

𝑡 𝑑𝑡

𝑇_𝑏 0

= 𝑠

𝑡 𝑑𝑡

𝑇_𝑏 0

= 𝐴

𝑇

2

Two-path correlation

receiver (general case)

𝑑𝑡

𝑇_𝑏 0

𝑑𝑡

𝑇_𝑏 0

×

×

Σ

+

−

Detection device

Choose 1 if 𝑙 > 0

Otherwise, choose 0 𝑥(𝑡)

x(t): received signal

𝑠₁(𝑡)

𝑠₀(𝑡)

Correlator: see how similar 𝑥 𝑡 and 𝑠₁(𝑡) are

Correlator: see how similar 𝑥 𝑡 and 𝑠₀(𝑡) are 𝑙

Coherent Detection

• 𝒘(𝒕): AWGN, 𝑵 𝟎,

• 𝑯

: 𝒙 𝒕 = 𝒔

𝒕 + 𝒘(𝒕)

• 𝑯

: 𝒙 𝒕 = 𝒔

𝒕 + 𝒘(𝒕)

• Receiver output:

• Decision level: 0

• If 𝑙 is larger than 1, than 𝑥(𝑡) is “more similar” to 𝑠

(𝑡)

• If 𝑙 is smaller than 1, than 𝑥(𝑡) is “more similar” to 𝑠

(𝑡)

𝑙 = 𝑥 𝑡 𝑠

𝑡 − 𝑠

𝑡 𝑑𝑡

𝑇_𝑏 0

Coherent Detection

• 𝑯

:

• Since the noise w(t) is zero-mean,

• 𝝆: the correlation coefficient of the signals 𝒔

(𝒕) and 𝒔

𝒕

𝑙 = 𝑠

𝑡 𝑠

𝑡 − 𝑠

𝑡 𝑑𝑡

0

− 𝑤 𝑡 𝑠

𝑡 − 𝑠

𝑡 𝑑𝑡

0

𝑙 = 𝑥 𝑡 𝑠

𝑡 − 𝑠

𝑡 𝑑𝑡

𝑇_𝑏 0

𝐸 𝐿 𝐻₁ = 𝑠₁ 𝑡 𝑠₁ 𝑡 − 𝑠₀ 𝑡 𝑑𝑡

𝑇_𝑏 0

= 𝐸_𝑏(1 − 𝜌)

𝜌 = 𝑠₀^𝑇𝑏 ₀ 𝑡 𝑠₁ 𝑡 𝑑𝑡 𝑠₀^𝑇𝑏 ₀² 𝑡 𝑑𝑡 𝑠₀^𝑇𝑏 ₁² 𝑡 𝑑𝑡

12

= 1

𝐸_𝑏 𝑠₀ 𝑡 𝑠₁ 𝑡 𝑑𝑡⁡

𝑇_𝑏 0

L: the random variable whose value is 𝑙

0 ≤ 𝜌 ≤ 1

Coherent Detection

• Similarly,

• L’s variance is the same for 𝑯

and 𝑯

. Since 𝒔

(𝒕) and 𝒔

(𝒕) is deterministic given the transmitted bit, we have

𝐸 𝐿 𝐻₀ = −𝐸_𝑏 1 − 𝜌

𝑉𝑎𝑟 𝐿 = E L − E L ²

= E 𝑤 𝑡 𝑤 𝑢 𝑠₁ 𝑡 − 𝑠₀ 𝑡 𝑠₁ 𝑢 − 𝑠₀ 𝑢 𝑑𝑡

𝑇_𝑏

0 𝑑𝑢

𝑇_𝑏 0

= 𝑬 𝒘 𝒕 𝒘 𝒖 𝑠₁ 𝑡 − 𝑠₀ 𝑡 𝑠₁ 𝑢 − 𝑠₀ 𝑢 𝑑𝑡

𝑇_𝑏

0 𝑑𝑢

𝑇_𝑏 0

= 𝛿(𝑡 − 𝑢) 𝑠₁ 𝑡 − 𝑠₀ 𝑡 𝑠₁ 𝑢 − 𝑠₀ 𝑢 𝑑𝑡

𝑇_𝑏

0 𝑑𝑢

𝑇_𝑏 0

= 𝛿(𝑡 − 𝑢) 𝑠₁ 𝑡 − 𝑠₀ 𝑡 𝑠₁ 𝑢 − 𝑠₀ 𝑢 𝑑𝑡

𝑇_𝑏

0 𝑑𝑢

𝑇_𝑏 0

= 𝑁₀

2 𝑠₁ 𝑡 − 𝑠₀ 𝑡 ²𝑑𝑡

𝑇_𝑏

0 = 𝑁₀𝐸_𝑏(1 − 𝜌)

• Therefore, we know that L conditioned on 𝑯

is a Gaussian distributed random variable: 𝑵 𝑬

𝟏 − 𝝆 , 𝑵

𝑬

𝟏 − 𝝆

Q Function

• Q function is defined over the CDF of Gaussian distribution 𝑵(𝟎, 𝟏)

𝑄 𝑥 = 1

2𝜋 exp − 𝑢

2 𝑑𝑢

∞

𝑥

= 1 − Φ(𝑥)

CDF of 𝑵(𝟎, 𝟏)

N(0,1)’s PDF

u f(u)

Integration (Area under the curve)  x

Bit Error Rate

𝑳|𝑯_𝟏~𝑵 𝑬_𝒃 𝟏 − 𝝆 , 𝑵_𝟎𝑬_𝒃 𝟏 − 𝝆

𝑬_𝒃 𝟏 − 𝝆

𝝇^𝟐~𝑵_𝟎𝑬_𝒃 𝟏 − 𝝆

Integration (Area under the curve) 0

How to express this area with Q function?

−𝑬_𝒃 𝟏 − 𝝆 0 Shift left by 𝑬_𝒃 𝟏 − 𝝆

−𝑬_𝒃 𝟏 − 𝝆 𝑵_𝟎𝑬_𝒃 𝟏 − 𝝆

0

Divide u by 𝑁₀𝐸_𝑏 1 − 𝜌

𝑵 𝟎, 𝑵_𝟎𝑬_𝒃 𝟏 − 𝝆 𝑵(𝟎, 𝟏)

Bit Error Rate

−𝑬_𝒃 𝟏 − 𝝆 𝑵_𝟎𝑬_𝒃 𝟏 − 𝝆

0

𝑵(𝟎, 𝟏)

𝑬_𝒃 𝟏 − 𝝆 𝑵_𝟎𝑬_𝒃 𝟏 − 𝝆 0

𝑵(𝟎, 𝟏)

𝑃

= 𝑄 𝐸

1 − 𝜌 𝑁

𝑃

= 𝑄 2𝐸

𝑁

𝑃

= 𝑄 𝐸

𝑁

For BPSK, 𝜌 = −1 For BFSK, 𝜌 = 0

Signal Space - BPSK

Inphase Quadrature

sin⁡(2𝜋𝑓_𝑐𝑡)

cos⁡(2𝜋𝑓_𝑐𝑡)

−𝐴_𝑐 𝐴_𝑐

Bit 0 Bit 1

𝑠₁ 𝑡 = 𝐴_𝑐cos⁡(2𝜋𝑓_𝑐𝑡) 𝑠₀ 𝑡 = −𝐴_𝑐cos⁡(2𝜋𝑓_𝑐𝑡) Noise

Energy

Signal Space - QPSK

Inphase Quadrature

sin⁡(2𝜋𝑓_𝑐𝑡)

cos⁡(2𝜋𝑓_𝑐𝑡)

Bit 01 Bit 11

𝑠₁₁ 𝑡 = 𝐴_𝑐 cos 2𝜋𝑓_𝑐𝑡 + 𝜋 4

𝑠₀₀ 𝑡 = 𝐴_𝑐 cos 2𝜋𝑓_𝑐𝑡 + 5𝜋 4

Bit 10 Bit 00

𝐴_𝑐 2

−𝐴_𝑐 2

−𝐴_𝑐 2

𝐴_𝑐 2 𝐴_𝑐

𝑠₀₁ 𝑡 = 𝐴_𝑐 cos 2𝜋𝑓_𝑐𝑡 + 3𝜋 4

𝑠₁₀ 𝑡 = 𝐴_𝑐 cos 2𝜋𝑓_𝑐𝑡 +7𝜋 4 𝜙

M-ary Modulation

Inphase Quadrature

𝐴_𝑐

𝜙 8-PSK

Inphase Quadrature

4-PSK

Increasing M would increase the data rate (given the same signal bandwidth)

M-ary Modulation

Inphase Quadrature

16-QAM

M-PAM BER versus SNR

𝐵𝐸𝑅 ≤ 𝑃_𝑒 ≤ 2 × 1 − 1

𝑀 × 𝑄 3 × 𝑆𝑁𝑅 𝑀² − 1

𝑆𝑁𝑅 = 𝐸_𝑏 𝜍_𝑛²

M-QAM BER versus SNR

𝐵𝐸𝑅 ≤ 𝑃_𝑒 ≤ 4 × 1 − 1

𝑀 × 𝑄 3 × 𝑆𝑁𝑅 𝑀 − 1

𝐵𝐸𝑅 ≤ 𝑃_𝑒 ≤ 4 × 1 − 1

2𝑀 × 𝑄 3 × 𝑆𝑁𝑅 31 × 𝑀32 − 1

M-PSK BER versus SNR

𝐵𝐸𝑅 ≤ 𝑃_𝑒 ≤ 2 × 𝑄 2 × 𝑆𝑁𝑅 × sin 𝜋 𝑀