ECE 342 Communication Theory Fall 2005, Class Notes Prof. Tiffany J Li www: http://www.eecs.lehigh.edu/∼jingli/teach email: [email protected]

Analog Communications

Modulation and Communication Systems • Modulation is a process that causes a shift in the range of frequencies in a signal. In effect, modulation converts the message signal from lowpass to bandpass. • Two types of communication systems. – Baseband communication: does not use modulation. ∗ The term baseband is used to designate the band of frequen- cies of the signal delivered by the source or the input transducer. – In telephony, the baseband is the audio band (band of voice signals) of 0 to 3.5kHz. – In television, the baseband is the video band occupying 0 to 4.3MHz. – For digital data or pulse-code modulation (PCM) using bipo- lar signaling at a rate Rb pulses per second, the baseband is 0 to Rb Hz. ∗ Baseband signals cannot be transmitted over a radio link but are suitable for transmission over a pair of wires, coaxial cables, or optical fibers. Examples include: – local telephony communication – short-haul PCM between two exchanges – long-distance PCM over optical fibers – Carrier communication: uses modulation. ∗ Eg: amplitude modulation, frequency modulation, phase mod- ulation.

1 ∗ A comment about pulse-modulated signals (including pulse amplitude modulation (PAM), pulse width modulation (PWM), pulse position modulation (PPM), pulse code modulation (PCM) and delta modulation (DM)): despite the term modulation, these signals are baseband signals. Pulse-modulation schemes are really baseband coding schemes, and they yield baseband signals. These signal must still modulate a carrier in order to shift their spectra.

Objectives of Modulation

• To translate the frequency of the lowpass signal to the passband of the channel so that the spectrum of the transmitted bandpass signal will match the passband characteristics of the channel. (Eg. in transmission of speech over microwave links, the transmission frequencies must be increased to the gigahertz range.) • To reduce the size of the antennas. (To obtain efficient radiation of electromagnetic energy, the antenna must be longer than 1/10 of the wavelength.) • To accommodate for the simultaneous transmission of signals from sev- eral message sources (e.g. frequency-division multiplexing). • To expand the bandwidth of the transmitted signal in order to increase its noise and interference immunity in transmission over a noisy channel (i.e. use modulation to exchange transmission bandwidth for the signal- to-noise ratio).

Energy Spectral Density and Power Spectral Density

• Three ways of computing the energy of a signal g(t):

∞ 2 ∞ 2 Eg = |g(t)| dt = Rg(0) = |G(f)| df Z−∞ Z−∞ ∗ where Rg(τ) = g(τ) ∗ g (−τ) is the (time) auto correlation function of g(t).

2 ∆ 2 • Gg(f) = |G(f)| is called the energy spectral density of a signal g(t). – |G(f)|2 equals the Fourier Transform of the time autocorrelation ∗ ∞ ∗ function Rg(τ) = g(τ) ∗ g (−τ) = −∞ g(t)g (t − τ)dt. R – It represents the amount of energy per unit bandwidth present in the signal at various frequencies. – Energy of the signal can be computed by integrating the energy spectral density over all frequencies: ∞ Eg = Gg(f)df Z−∞ – If g(t) is passed through a filter/channel with the impulse response h(t), and the output signal is y(t), then the energy of the output signal is ∞ 2 2 Ey(t) = |G(f)| |Y (f)| df Z−∞ and the energy spectral density of the output signal y(t) is

2 2 Gy(f) = Gg(f)Gh(f) = |G(f)| |H(f)|

∆ 1 2 • Sg(f) = limT →∞ T |G(f)| is called the power spectral density (PSD) of a signal g(t)

– Sg(f) equals the Fourier Transform of the time-average autocorre- 1 T/2 ∗ lation function Rg(τ) = limT →∞ T −T/2 g(t)g (t − τ)dt. R – It represents the amount of power per unit bandwidth present in the signal at various frequencies. – The power of the signal can be computed by integrating the PSD over all frequencies: ∞ Pg = Sg(f)df Z−∞ – If g(t) is passed through a filter/channel with the impulse response h(t), and the output signal is y(t), then the power of the output signal is 1 T/2 2 2 Py(t) = lim |G(f)| |Y (f)| df T →∞ T Z−T/2 and the energy spectral density of the output signal y(t) is

Sy(f) = Sg(f)Sh(f)

3 Modulation

• Consider a lowpass, power-type signal, m(t), of bandwidth, W (that is, 1 T/2 2 M(f) = 0, for |f| > W ). Let Pm = limT →∞ T −T/2 |m(t)| dt denote the R power of the signal. The message signal m(t) is transmitted through the communication channel by impressing it on a sinusoidal carrier signal of the form c(t) = Accos(2πfct + φc)

where Ac is the carrier amplitude, fc is the carrier frequency, φc is the carrier phase. • We say that the message signal m(t) modulates the carrier signal c(t) in either amplitude, frequency, or phase if after modulation, the ampli- tude, frequency, or phase of the signal become functions of the message signal. This results in amplitude modulation (AM), frequency modulation (FM), or phase modulation (PM). The latter two types of modulation are similar, and belong to the class of modulation known as angle modulation.

• Four types of amplitude modulation:

– Double-Sideband Suppressed-Carrier AM (DSB-SC) – Conventional Double-Sideband AM (DSB, a.k.a. conventional AM) – Single-Sideband AM (SSB) – Vestigial-Sideband AM (VSB)

4 Double-Sideband Suppressed-Carrier AM

• lowpass message signal m(t), bandwidth W ; also assume the average of m(t) is zero (which is a valid assumption for many signals including audio signals).

• carrier signal c(t) = Accos(2πfct) (without loss of generality, we as- sumed the carrier phase is 0) • DSB-SC AM modulated signal

u(t) = m(t)c(t) = Acm(t)cos(2πfct)

– u(t) has bandwidth 2W , i.e. double the bandwidth of m(t).

– u(t) has double sidebands: upper sideband |f| > fc; lower side- band |f| < fc. – u(t) does not contain a carrier component (hence carrier suppressed). All the transmitted power is contained in the modulating (message) signal m(t). • Spectrum of DSB-SC signals: Assume m(t) ⇐⇒ M(f), A U(f) = F[A m(t)cos(2πf t)] = c [M(f − f ) + M(f + f )] c c 2 c c • Power content of DSB-SC signals

1 T/2 2 Pu = lim u (t)dt T →∞ T Z−T/2 1 T/2 2 t 2 = lim Acm (t)cos (2πfct)dt T →∞ T Z−T/2 2 2 Ac 1 T/2 2 Ac 1 T/2 2 = lim m (t)dt + lim m (t)cos(4πfct)dt 2 T →∞ T Z−T/2 2 T →∞ T Z−T/2 A2 = c P (1) 2 m 2 (The overall integral of m (t)cos(4πfct) is almost zero. Since the result of the integral is divided by T , and T becomes very large, the second term in the equation becomes zero.) • Demodulation of DSB-SC signals:

5 – Requires a phase-coherent or synchronous demodulator. That is, the phase φ of the locally generated sinusoid should ideally be equal to the phase of the received-carrier signal. – Steps (synchronous demodulator): ∗ STEP 1. first multiply received signal r(t) by a locally gener- ated sinusoid cos(2πfct + φ) ∗ STEP 2. pass the product signal through an ideal lowpass filter with the bandwidth W ∗ Mathematically:

r(t) = u(t) = Acm(t)cos(2πfct) assuming noiseless channel

r(t)cos(2πfct + φ) = Acm(t)cos(2πfct)cos(2πfct + φ) 1 1 = A m(t)cos(φ) + A m(t)cos(4πf t + φ) (2) 2 c 2 c c filtered out | {z } The power in the demodulated signal is decreased by a factor of cos2φ =⇒ in need for a synchronous demodulator

• Two ways to generate phase-locked sinusoidal signals at the receiver: Phase-locked loop (PLL) and using a pilot tone.

• The method of pilot tone:

– Add a pilot tone, i.e. a carrier component, into the transmitted signal. 2 – The power of the pilot tone, Ap/2 is selected to be significantly smaller than that of the modulated signal u(t). – The transmitted signal is still double sideband, but no longer car- rier suppressed.

– At the receiver, a narrow band filter tuned to frequency fc filters out the pilot signal component; its output is used to multiply the received signal. – Note that the presence of the pilot signal results in a DC component in the demodulated signal; this needs to be subtracted out in order to recover m(t).

6 – Disadvantages of pilot tone: a certain portion of the transmitted signal power is allocated to the transmission of the pilot.

7 Conventional DSB AM

• A conventional AM modulated signal consists of a large carrier compo- nent, in addition to the double-sideband AM modulated signal:

u(t) = Ac[1 + m(t)]cos(2πfct)

where the message signal is constrained to satisfy |m(t)| ≤ 1. (If m(t) < −1, the AM signal is over-modulated, causing a more com- plex demodulation process.)

• For convenience, let us express m(t) as

m(t) = amn(t)

where mn(t) = m(t)/ max |m(t)|, and a is called the modulation in- dex and 0 < a < 1. Hence

u(t) = Ac[1 + amn(t)]cos(2πfct)

• Spectrum of the conventional AM signal:

U(f) = F[Ac[1 + amn(t)]cos(2πfct)] A a A = c [M (f − f ) + M (f + f )] + c [δ(f − f ) + δ(f + f )] 2 n c n c 2 c c

• Power of the conventional AM signal: A2 A2 P = c (1 + P ) = c (1 + a2P ) u 2 m 2 mn

• Demodulation:

– No need for a synchronous demodulator. – We can use a simple envelope detector (i.e. a rectifier + lowpass filter) followed by a transformer to demodulate it: ∗ STEP 1. Rectify the signal: i.e. eliminating the negative val- ues. This would not affect the message signal since the envelop (amplitude) 1 + m(t) > 0.

8 ∗ STEP 2. Pass the rectified signal through a low pass filter whose bandwidth matches that of the message signal.

∗ STEP 3. Eliminate the DC component in the output of the envelope detector through a transformer. Note the output of the envelope detector (i.e. a rectifier and a lowpass filter) is (ideally) d(t) = g1 +g2m(t), where g1 is the DC term (introduced by the carrier component at the transmitter) and g2 is a gain factor due to the signal demodulator. • Power efficiency:

2 useful power Pm a Pmn η = = = 2 total power 1 + Pm 1 + a Pmn For practical signals, the efficiency is on the order of 25% or lower. • Major advantage of the conventional AM is the simple demodulator =⇒ widely used in AM radio broadcast.

9 Single Sideband (SSB) AM

• The DSB spectrum has two sideband: upper side-band (USB) and low side-band (LSB), both containing the complete information of the base- band signal. =⇒ sufficient to transmit only one sideband: Single-sideband trans- mission. SSB transmission requires only one-half the bandwidth of the DSB signal. • The Single-sideband (suppressed-carrier) signals can be demodulated using a synchronous demodulator in a way identical to demodulating DSB-SC signals:

– Multiplying a USB (LSB) signal by cos(2πfct) shifts its spectrum to the left and right by fc. – Low-passing filtering of the signal yields the desired baseband sig- nal. • Generating SSB AM signals: – In the frequency domain: generate double-sideband signals and use an ideal bandpass filter to filter one of the sidebands of a DSB signal. – In the time domain:

Upper sideband: u(t) = Acm(t)cos(2πfct) − Acmˆ (t)sin(2πfct)

Lower sideband: u(t) = Acm(t)cos(2πfct) + Acmˆ (t)sin(2πfct)

where mˆ (t) is the Hilbert transform of m(t). • Hilbert transform of a signal m(t) results in a new signal mˆ(t) that has exactly the same frequency components present in m(t) with the same amplitude - except there is a 90o phase delay (i.e. a π/2 phase shifter). – In effect, after Hilbert transform, the spectrum of the signal at posi- tive frequencies is multiplied by −j (i.e. ej2πf0t becomes ej2πf0t−π/2), and the spectrum of the signal at negative frequencies is multiplied by +j (i.e. e−j2πf0t becomes e−j(2πf0t−π/2))

10 – This is equivalent to multiplying the spectrum (Fourier transform) of the signal by −j sgn(f) (or filtering by a filter with response −j sgn(f)): if m(t) ⇐⇒ M(f) then mˆ (t) ⇐⇒ −j sgn(f)M(f)

1 – Since πt ⇐⇒ −jsgn(f), in the time domain, the operation of Hilbert transform is equivalent to a convolutional operation:

1 1 ∞ m(τ mˆ (t) = ∗ m(t) = dτ πt π Z−∞ t − τ • The impact of a phase-offset in the demodulator:

– Consider for example USB signals:

r(t) = m(t) = Acm(t)cos(2πfct) − Acmˆ (t)sin(2πfct)

1 1 r(t)cos(2πf t + φ) = A m(t)cos(φ) + mˆ (t)sin(φ) c 2 c 2 +double frequency terms

– The impact of phase-offset is two-fold: - It reduces the amplitude of of the desired signal m(t) by cos(φ) (same as DSB-SC) - It also introduces an undesirable sideband signal (interference) due to the presence of mˆ (t) – Hence, coherent demodulator is desired, and the use of pilot tone is a very effective method in SSB Am systems (at the cost of some power efficiency).

11 Vestigial Sideband (VSB) AM

• Motivation:

– SSB AM is bandwidth efficient. The filter method to generate SSB requires an exact sideband filter which has an extremely sharp cutoff in the vicinity of the carrier (particularly when m(t) has a large power concentrated in the vicinity of f = 0). Such filter characteristics are very difficult to implement in practice. – Relax the SSB system by allow vestige, which is a portion of the unwanted sideband.

• Justification:

– VSB AM system can be implemented by generating a DSB-SC AM signal and passing it through a sideband filter with the frequency response H(f). – To recover the original message spectrum, the VSB-filter charac- teristics must satisfy:

H(f − fc) + H(f + fc) = constant for any |f| ≤ W

where W is the bandwidth of the message signal m(t). – In practice, to minimize distortion of the message signal, the VSB- filter should also have a linear phase over its passband.

• Practice:

– Suitable for signals that have a strong low-frequency component, such as video signals. VSB used in standard TV broadcasting. – A carrier component is generally transmitted so that the demodula- tor can extract a phase-coherent reference for synchronous demod- ulation. In TV broadcast, a large carrier component is transmitted. and it is possible to recover the message using a simple envelope detector (same as the conventional DSB AM in radio broadcast- ing).

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