Quadrature Amplitude Modulation (QAM)

ò PAM signals occupy twice the bandwidth required for the baseband ò Transmit two PAM signals using carriers of the same frequency but in phase and quadrature

ò Demodulation: QAM Transmitter Implementation

ò QAM is widely used method for transmitting digital data over bandpass channels ò A basic QAM transmitter Inphase component Pulse a Impulse a*(t) a(t) n Shaping Modulator + d Map to Filter sin w t n Serial to c Constellation g (t) s(t) Parallel T Point * Pulse 1 Impulse b (t) b(t) Shaping - J Modulator bn Filter quadrature component gT(t) cos wct

ò Using complex notation

= − = ℜ s(t) a(t)cos wct b(t)sin wct s(t) {s+ (t)} which is the preenvelope of QAM ∞ ∞ − = + − jwct = jwckT − jwc (t kT ) s+ (t) ƒ(ak jbk )gT (t kT )e s+ (t) ƒ(ck e )gT (t kT)e −∞ −∞ Complex Symbols

ò Quadriphase-Shift Keying (QPSK) ò 4-QAM ò Transmitted signal is contained in the phase

imaginary imaginary

real real M-ary PSK

ò QPSK is a special case of M-ary PSK, where the phase of the carrier takes on one of M possible values

imaginary

real Pulse Shaping Filters

ò The real pulse shaping g (t) filter response T

= jwct = + ò Let h(t) gT (t)e hI (t) jhQ (t) = hI (t) gT (t)cos wct = where hQ (t) gT (t)sin wct

= − ò This filter is a bandpass H (w) GT (w wc ) filter with the frequency response Transmitted Sequences

ò Modulated sequences before passband shaping

' = ℜ jwc kT = − ak {ck e } ak cos wc kT bk sin wc kT

' = ℑ jwc kT = + bk {ck e } ak sin wc kT bk cos wc kT

ò Using these definitions

∞ = ℜ = ' − − ' − s(t) {s+ (t)} ƒ ak hI (t kT) bk hQ (t kT) k =−∞

ò Advantage: computational savings Ideal QAM Demodulation

ò Exact knowledge of the carrier and symbol clock phases and frequencies ò Apply the Hilbert Transform of the received signal to

generate the pre-envelope s+(t) ∞ − = jwct = + − s(t) s+ (t)e ƒ (ak jbk )gT (t kT) k =−∞

ò If gT(t) has no intersymbol interference, we get exactly the transmitted symbol = + s"(nT ) an jbn QAM Demodulation

ò QAM demodulator using the complex envelope

e-jwct

s(t) Receive Filter

Hilbert Transform Filter Second Method

ò Based on a pair of DSBSC-AM coherent demodulators a(t) LPF s(t) 2coswct

b(t) LPF

-2sinwct ò In DSP implementations, Hilbert transform method is popular because ò It only requires single filter

ò 2wc terms are automatically cancelled QAM Front-End Subsystems

ò Automatic gain control ò Full scale usage of A/D and avoid clipping ò No need to implement AGC in your project ò The Carrier detect subsystem ò Determine if a QAM signal is present at the receiver ò This power estimate can be easily formed by averaging the squared ADC output samples r(n):

ò You can select c as a number slightly less than 1, i.e, 0.9. ò When the power estimate p(n) exceeds a predetermined threshold for a period of time, a received QAM signal is declared to be present. Carrier Detect

ò Determine a threshold value by inspecting your power estimate. ò The following figure shows the portion of received QAM signal and the power estimate average. Frame Synchronization

ò Assume that the receiver knows marker symbols (say, 10 complex symbols, m(n)) ò Assume that s(k) is the output of your QAM demodulator. Note that s(k) and m(n) are complex. ò Of course, in DSP you define complex numbers by keeping real and imaginary parts in different buffers. Frame Synchronization

ò Correlation, ò Peak=

ò Peakindex=max(abs(Peak)) ò Note that Peak is complex quantity Initial Phase Correction

θ ò Defining ej =peak/abs(peak), ò then, the phase shift on s(n) which is the output of QAM demodulator can be corrected as θ ò x(k)=s(index+k-1)e-j Switch to QAM Modulation

ò You can proceed with hilbert transform filtering and complex downconversion by setting your receiver pointer to the beginning of your input buffer. ò First the receiver is on and transmitter is off ò Transmitter starts with marker and PN sequence, both complex ò When transmitter starts, carrier detect will switch to QAM downconversion Carrier Offset Impairment for QAM

ò Received QAM signal

ò After Hilbert transform, we can form

π φ ò Complex demodulator ej2 fct+j )

ò Obtain m1(t) as the real part Carrier Offset Impairment for QAM

ò For PAM signaling carrier offset causes scaling with cosine of the phase offset ò For QAM signaling carrier offset causes rotation of the constellation by the carrier offset

θ-φ