Chapter 9. Amplitude Modulation

Chapter 9. Amplitude Modulation

1

Goals

 Modulate and demodulate the double sideband-suppressed carrier (DSB-SC) amplitude modulation (AM) signals.

 Investigate the effects of frequency and phase errors on the demodulation performance.

 Modulate and demodulate AM signals by using the sampling and band pass filter (BPF) technique.

2

Double side-band suppressed carrier(DSB-SC)

3

𝑓𝑓 𝑡𝑡

cos 𝜔𝜔

𝑡𝑡

∅(𝑡𝑡)=𝑓𝑓 𝑡𝑡 × cos(𝜔𝜔

𝑡𝑡)

 Amplitude modulations(AM)

 DSB-SC, DSB-LC, SSB, VSB, etc.

 DSB-SC

 Modulated signal ∅(𝑡𝑡)=𝑓𝑓 𝑡𝑡 cos(𝜔𝜔

𝑡𝑡)

• 𝑓𝑓 𝑡𝑡 : information signal

• cos 𝜔𝜔

𝑡𝑡 : carrier signal, 𝜔𝜔

= carrier frequency.

 Transmitter(modulator) structure

 Spectrum of DSB-SC

F ∅ 𝑡𝑡 = F ∅ 𝑡𝑡 = F 𝑓𝑓 𝑡𝑡 cos(𝜔𝜔

𝑡𝑡) = 1

2 𝐹𝐹 𝜔𝜔 + 𝜔𝜔

+ 1

2 𝐹𝐹 𝜔𝜔 − 𝜔𝜔

• Recall that multiplying sinusoid = frequency shift

Double side-band suppressed carrier(DSB-SC)

 DSB-SC

 Receiver (demodulator) structure

 [CH19.2B9] Proof:

𝑔𝑔 𝑡𝑡 = ∅ 𝑡𝑡 cos 𝜔𝜔

𝑡𝑡 = 𝑓𝑓 𝑡𝑡 cos(𝜔𝜔

𝑡𝑡)

= 𝑓𝑓(𝑡𝑡)

=

2

𝑓𝑓 𝑡𝑡 +

𝑓𝑓 𝑡𝑡 cos(2𝜔𝜔

𝑡𝑡)

𝐺𝐺(𝜔𝜔)=

𝐹𝐹 𝜔𝜔 +

𝐹𝐹 𝜔𝜔 − 2𝜔𝜔

+

𝐹𝐹 𝜔𝜔 + 2𝜔𝜔

• Here, let us define LPF[x]: LPF[x] ≜ Ideal(Perfectly flat magnitude response within BW, no delay) LPF output of x.

Then,

LPF[ 𝐺𝐺(𝜔𝜔)]=

𝐹𝐹 𝜔𝜔 +

𝐹𝐹 𝜔𝜔 − 2𝜔𝜔

+

𝐹𝐹 𝜔𝜔 + 2𝜔𝜔

=

2

𝐹𝐹 𝜔𝜔

 LPF 𝑔𝑔 𝑡𝑡 = LPF

𝑓𝑓 𝑡𝑡 +

𝑓𝑓 𝑡𝑡 cos(2𝜔𝜔

𝑡𝑡) =

𝑓𝑓(𝑡𝑡)

∅(𝑡𝑡)=𝑓𝑓 𝑡𝑡 cos(𝜔𝜔

𝑡𝑡)

𝑓𝑓 𝑡𝑡 : demodulated signal

cos 𝜔𝜔

𝑡𝑡 :local carrier signal

𝑔𝑔(𝑡𝑡)

Waveforms and Spectra of DSB-SC

 [CH9.1F4~1F6]If the BW of LPF is larger than W, noise power in the demodulated signal increases.

 [CH9.1F4~1F6]If the BW of LPF is smaller than W, the demodulated information is distorted. ⁵

LPF

cos 𝜔𝜔𝑐𝑐𝑡𝑡

Effect of phase error

 What if there exists a phase error between the carrier and the local carrier?

Say, local carrier = cos 𝜔𝜔

𝑡𝑡 +

not cos 𝜔𝜔

𝑡𝑡 . Then,

 𝑔𝑔 𝑡𝑡 = ∅ 𝑡𝑡 cos 𝜔𝜔

𝑡𝑡 +

= 𝑓𝑓 𝑡𝑡 cos(𝜔𝜔

𝑡𝑡)cos 𝜔𝜔

𝑡𝑡 +

= 𝑓𝑓(𝑡𝑡)

cos 2𝜔𝜔

𝑡𝑡 +

+ cos −𝜃𝜃

=

𝑓𝑓 𝑡𝑡 cos 2𝜔𝜔

𝑡𝑡 +

+

𝑓𝑓 𝑡𝑡 cos

 [CH9.2A4] the demodulated signal(LPF output in the receiver) is given as:

LPF 𝑔𝑔 𝑡𝑡 = LPF 1

2 𝑓𝑓(𝑡𝑡) cos 2𝜔𝜔

𝑡𝑡 +

+ 1

2 𝑓𝑓(𝑡𝑡)cos

=LPF

𝑓𝑓(𝑡𝑡) cos 2𝜔𝜔

𝑡𝑡 +

+LPF

𝑓𝑓(𝑡𝑡)cos

=

𝑓𝑓 𝑡𝑡 cos(𝜃𝜃)

 Phase error results in decreased amplitude of the demodulated signal by the factor of cos(𝜃𝜃).

Special case: if

=

2

, then the signal disappears.

6

Effect of phase error

 [CH9.2B]Consider the background noise 𝑛𝑛 𝑡𝑡 is added to the received signal 𝑓𝑓 𝑡𝑡 cos(𝜔𝜔_𝑐𝑐𝑡𝑡):

 Then, the multiplier output in the receiver 𝑔𝑔 𝑡𝑡 = 𝑓𝑓 𝑡𝑡 cos 𝜔𝜔_𝑐𝑐𝑡𝑡 + 𝑛𝑛(𝑡𝑡) cos 𝜔𝜔_𝑐𝑐𝑡𝑡 +𝜃𝜃

= 𝑓𝑓 𝑡𝑡 cos 𝜔𝜔_𝑐𝑐𝑡𝑡 cos 𝜔𝜔_𝑐𝑐𝑡𝑡 +𝜃𝜃 + 𝑛𝑛(𝑡𝑡) cos 𝜔𝜔_𝑐𝑐𝑡𝑡 +𝜃𝜃

 𝑛𝑛 𝑡𝑡 cos 𝜔𝜔_𝑐𝑐𝑡𝑡 + 𝜃𝜃 = the noise component in 𝑔𝑔 𝑡𝑡 . We denote 𝑛𝑛_𝜃𝜃 𝑡𝑡 = 𝑛𝑛 𝑡𝑡 cos 𝜔𝜔_𝑐𝑐𝑡𝑡 + 𝜃𝜃

.

 [CH9.2B10a] ACF of 𝑛𝑛_𝜃𝜃 𝑡𝑡

 IF 𝜏𝜏=0, 𝑅𝑅 _{𝑛𝑛𝜃𝜃} 𝜏𝜏 = 0 = lim_{𝑇𝑇→∞}^{∫− ⁄𝑇𝑇 2}

⁄

𝑇𝑇 2 𝑛𝑛 𝑡𝑡 cos 𝜔𝜔𝑐𝑐𝑡𝑡+𝜃𝜃 ^∗𝑛𝑛 𝑡𝑡 cos 𝜔𝜔𝑐𝑐𝑡𝑡+𝜃𝜃 𝑑𝑑𝑡𝑡

𝑇𝑇 =

𝑇𝑇→∞lim

∫_{− ⁄}^{𝑇𝑇 2}_{𝑇𝑇 2}^⁄ 𝑛𝑛 𝑡𝑡 cos 𝜔𝜔_𝑐𝑐𝑡𝑡+𝜃𝜃 ²𝑑𝑑𝑡𝑡

𝑇𝑇 =Power of 𝑛𝑛_𝜃𝜃 𝑡𝑡

• Two sample cases of 𝑛𝑛_𝜃𝜃 𝑡𝑡

• These samples illustrate that Power of 𝑛𝑛_𝜃𝜃 𝑡𝑡 = Power of 𝑛𝑛 𝑡𝑡 × Power of cos(𝜔𝜔𝑡𝑡+𝜃𝜃), where power of cos(𝜔𝜔𝑡𝑡+𝜃𝜃) =1/2 irrespective of 𝜃𝜃 .

 𝑅𝑅_{𝑛𝑛𝜃𝜃} 𝜏𝜏 = 0 =0.5 𝑅𝑅_𝑛𝑛 𝜏𝜏 = 0 ⁷

𝜃𝜃 = 0 𝜃𝜃 = 𝜋𝜋/3

𝑛𝑛 𝑡𝑡 cos 𝜔𝜔_𝑐𝑐𝑡𝑡 + 𝜃𝜃 𝑛𝑛_𝜃𝜃 𝑡𝑡 = 𝑛𝑛 𝑡𝑡 cos 𝜔𝜔_𝑐𝑐𝑡𝑡 + 𝜃𝜃

Effect of phase error

 [CH9.2B10a] (continued) ACF of 𝑛𝑛

𝑡𝑡

 𝑅𝑅

𝜏𝜏 = Correlation between 𝑛𝑛 𝑡𝑡 cos 𝜔𝜔

𝑡𝑡 + 𝜃𝜃 and 𝑛𝑛 𝑡𝑡 + 𝜏𝜏 cos(𝜔𝜔

(𝑡𝑡 +

8

Effect of phase error

 Demodulated signal (LPF output)

= signal term + noise term

=

2

𝑓𝑓 𝑡𝑡 cos(𝜃𝜃) + LPF[𝑛𝑛

𝑡𝑡 ]

Signal amplitude decreases by the factor of cos 𝜃𝜃  Signal power decreases by the factor of cos

𝜃𝜃

On the other hand, noise power is constant irrespective of 𝜃𝜃

 [CH9.2B4] Signal to Noise power Ratio (SNR) SNR (𝜃𝜃) =

Noise power

=

1

2𝑓𝑓 𝑡𝑡 cos(𝜃𝜃)

Power of LPF[𝑛𝑛_𝜃𝜃 𝑡𝑡 ]

=

1 2𝑓𝑓 𝑡𝑡 Power of LPF[𝑛𝑛_𝜃𝜃 𝑡𝑡 ]

= cos

𝜃𝜃 SNR(𝜃𝜃 = 0)

 Phase error results in SNR decrease by the factor of cos

𝜃𝜃

 [CH9.2B5] SNR loss = cos

𝜃𝜃 = 10𝑙𝑙𝑙𝑙𝑔𝑔

cos

𝜃𝜃 dB

9

Effect of frequency error

 Now we consider the case when

local carrier =cos 𝜔𝜔_𝑐𝑐 + ∆𝜔𝜔 𝑡𝑡 not cos 𝜔𝜔_𝑐𝑐𝑡𝑡 .

 We can rewrite

local carrier = cos 𝜔𝜔_𝑐𝑐𝑡𝑡 + ∆𝜔𝜔𝑡𝑡 =cos 𝜔𝜔_𝑐𝑐𝑡𝑡 + 𝜃𝜃 where 𝜃𝜃

=

.

• This implies that the frequency error results in linearly increasing phase error.

• [CH9.2C1] So, we only need to change 𝜃𝜃 into ∆𝜔𝜔𝑡𝑡 in all the derivations for the phase error case.

– For example : demodulated signal=¹

2𝑓𝑓 𝑡𝑡 cos(∆𝜔𝜔𝑡𝑡)

 So, Signal to Noise power Ratio (SNR)

SNR(∆𝜔𝜔𝑡𝑡) =cos² ∆𝜔𝜔𝑡𝑡 SNR ∆𝜔𝜔 = 0

• SNR is time varying according to cos² ∆𝜔𝜔𝑡𝑡 .

• [CH9.2C2,C3,C6,C7] Even with a very tiny frequency error, cos² ∆𝜔𝜔𝑡𝑡 periodically changes from 0 to 1.

• [CH9.2C2,C3,C6,C7] Whenever cos² ∆𝜔𝜔𝑡𝑡 approaches 0, SNR instantaneously becomes 0.  Critical problem!

10

Generating AM signal without Oscillator

 To generate carrier signal cos(𝜔𝜔_𝑐𝑐𝑡𝑡), we need an oscillator which is expensive.

 [CH9.3B] DSB-SC is equal to frequency shifting to center frequencies = ±𝜔𝜔_𝑐𝑐

 Recall, in chapter 7, frequency shifting is possible by sampling and BPF

technique if we properly set the sampling frequency and properly design BPF.

• BPF output spectrum=𝑃𝑃_−𝑛𝑛𝑋𝑋 𝜔𝜔 + 𝑛𝑛𝜔𝜔₀ + 𝑃𝑃_𝑛𝑛𝑋𝑋 𝜔𝜔 − 𝑛𝑛𝜔𝜔₀

• BPF output signal= 𝑓𝑓(𝑡𝑡)2 𝑃𝑃_𝑛𝑛 cos 𝑛𝑛𝜔𝜔₀𝑡𝑡 + 𝜃𝜃_{𝑃𝑃𝑛𝑛}

= DSB-SC modulated signal with a carrier signal =2 𝑃𝑃_𝑛𝑛 cos 𝑛𝑛𝜔𝜔₀𝑡𝑡 + 𝜃𝜃_{𝑃𝑃𝑛𝑛}

 [CH9.3B4] Parameter setting

Condition 1: 𝑛𝑛𝜔𝜔₀=carrier frequency 𝜔𝜔_𝑐𝑐, n=positive integer

Condition 2: The sampling frequency 𝜔𝜔₀ ≥ 2 ×Bandwidth of 𝑓𝑓 𝑡𝑡 .

• Possible sampling frequency 𝜔𝜔₀ ∶ 𝑆𝑆𝑆𝑆𝑙𝑙𝑆𝑆𝑆𝑆𝑡𝑡 𝑙𝑙𝑛𝑛𝑆𝑆 𝑙𝑙𝑓𝑓 𝜔𝜔_𝑐𝑐,^𝜔𝜔₂^𝑐𝑐,^𝜔𝜔₃^𝑐𝑐,^𝜔𝜔₄^𝑐𝑐, … under

the condition 2 ¹¹

P0

P1

P0

P1 _Pn −nω₀ −ω₀ ω0

nω0

P₋1

P−n

P0

P1 P_n

nω0

−

Sampling of 𝑓𝑓 𝑡𝑡

by a periodic signal 𝑝𝑝(𝑡𝑡)

BPF BPF output

=Spectrum of DSB-SC