Lecture 12 - Analog Communication (II)

Home , Analytic signal, Baseband, Passband, Pilot signal

Spring 2009

Colorado State University Dept of Electrical and Computer Engineering ECE423 – 1 / 12 Outline

● QAM: quadrature amplitude modulation ● SSB: single sideband ● Lab suggestions

Reference: ● Tretter Chapters 5-8 (On e-reserve), ● Haykin, Communication Systems, Third Edition

Colorado State University Dept of Electrical and Computer Engineering ECE423 – 2 / 12 QAM Basics

● QAM=”Quadrature Amplitude Modulation” ● QAM is both a generic term which encompasses AM, FM, DSB-SC, and SSB (”linear modulation techniques”) and a speciﬁc technique for sending two independent messages over one carrier. ● For AM and SSB with pilot signal, the receiver oscillator can be derived from s(t); for DSB-SC and SSB without pilot signal, the receiver oscillator must be independently created.

Colorado State University Dept of Electrical and Computer Engineering ECE423 – 3 / 12 QAM Modulator/Demodulator

● A picture of QAM commonly used in analog communication is given below. There is an alternate picture which is more useful and common in DSP, which will be shown later. ● For this representation, using the othogonality of sin(.) and cos(.), it is straightforward to show that the outputs of the LPF blocks are proportional to x(t) and y(t) as shown in the ﬁgure.

Colorado State University Dept of Electrical and Computer Engineering ECE423 – 4 / 12 Modulation Types

● For AM, x(t) = Ac[1 + kam(t)], y(t) = 0.

● For DSB-SC, x(t) = Acm(t), y(t) = 0.

● For SSB, x(t) = m(t), y(t) = m(ˆt). ● For QAM modulation, x(t) and y(t) can be independent signals, called in-phase and quadrature respectively.

● For FM, x(t) = cos(φ(t)), y(t) = sin(φ(t)).

Colorado State University Dept of Electrical and Computer Engineering ECE423 – 5 / 12 Complex Baseband Representation of Passband Signals

● We can picture x(t) and y(t) as being real and imaginary components of a complex baseband signal z(t) = x(t) + jy(t). ● z(t) can be transformed into a real passband signal and vice-versa. This is modulation and de-modulation. ● With independent x(t) and y(t), the modulated passband signal will be similar to DSB-SC in bandwidth utilization, but we will send two independent channels in the same bandwidth.

● If x(t) and y(t) are not independent, but instead y(t) = ±xˆ(t), ˆ≡ Hilbert tranform, then we can send one channel in either the upper sideband or lower sideband only (SSB), thus using half the bandwidth of DSB-SC and power of DSB-SC.

Colorado State University Dept of Electrical and Computer Engineering ECE423 – 6 / 12 Modulation of Complex Baseband Signal

jωct ● Let w(t) = z(t)e , where ωc is the carrier frequency. We know from the Modulation Theorem that W (ω) = Z(ω − ωc). W has only a positive frequency component (is an ”analytic signal”) and spans a bandwidth of 2B, which is the bandwidth of z(t). ● We can form a real passband signal by converting w(t) to real: 1 s(t) = 2 [w(t) + w(t)], where w(t) is the complex conjugate of w(t). Can jωct −jωct easily show that s(t) = z(t)e + z(t)e = x(t)cos(ωct) − y(t)sin(ωct).

Colorado State University Dept of Electrical and Computer Engineering ECE423 – 7 / 12 Demodulation of Complex Baseband Signal

● Assume the received signal is s(t) (no channel distortion).

● First put s(t) through a ”phase splitter” ﬁlter φ(t) ↔F T Φ(jω) = u(ω), where u(ω) is a frequency step function. The phase splitter discards the negative frequency term in s(t) and we recover w(t) = z(t)ejωct.

● Next multiply w(t) by a locally-generated complex exponential e−jωct to recover z(t).

Colorado State University Dept of Electrical and Computer Engineering ECE423 – 8 / 12 Recovering Carrier

We cannot regenerate the carrier using the quadrature component, as with the Costas receiver. There are two strategies for dealing with this: 1. Free-running at the carrier frequency, with some phase error (distortion). This is tolerable for many applications (voice).

2. Add small DC component to m(t) – ”pilot tone”. This requires that m(t) be passband (no DC component). Then s(t) = x(t)cos(ωct) − y(t)sin(ωct) + pcos(ωct). This produces a small carrier signal which can be locked onto by a PLL or ampliﬁed to create the ”local oscillator” signal.

Colorado State University Dept of Electrical and Computer Engineering ECE423 – 9 / 12 SSB Modulation

● To achieve SSB, the only difference is that we generate the complex baseband signal from a real baseband signal m(t) by passing through a phase splitter. This creates an analytic signal z(t) = m(t) ± jmˆ(t) which has only positive or negative frequency components. This is the positive or negative pre-envelope. When multiplied by ejωct and converted to real, s(t) will have only upper or lower sideband components, depending on the sign in z(t). ● De-modulation in principle is the same as before.

Colorado State University Dept of Electrical and Computer Engineering ECE423 – 10 / 12 Practical: Creating a Phase Splitter

● Remember that H(jω) = −jsgn(ω) is the Hilbert Transform. There are cof ﬁles for various orders of Hilbert Transformers in the lab support ﬁles. ● How do you multiply by j?

r(t) = s(t) + w(t) = s(t) + jǆ(t)

-j sgn(Z) j

Phase Splitter: )( jZ)

Colorado State University Dept of Electrical and Computer Engineering ECE423 – 11 / 12 Practical: QAM Receiver using Pilot Tone

Re[·] BPF DAC x(t) received signal x -j sgn(Ȧ)

nto Hilbert Transformer Im[·] BPF DAC y(t)

1 r Baseband Filters 1 2 2 That Remove DC 1 2rcos(Zcto )z r z Components

In-phase Carrier 1 1 r cos(Zcto )z

Quadrature Carrier 1 rsin(Zcto )z

Bandpass Carrier Recovery Filters

Colorado State University Dept of Electrical and Computer Engineering ECE423 – 12 / 12