Chapter 4 Bandpass Signalling Definitions Complex Envelope Representation

Chapter 4 Bandpass Signalling Ø Definitions Ø Complex Envelope Representation Ø Representation of Modulated Signals Ø Spectrum of Bandpass Signals Ø Power of Bandpass Signals Ø Examples Huseyin Bilgekul Eeng 360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University 1

Bandpass Signals Ø Energy spectrum of a bandpass signal is concentrated around the carrier frequency fc. Bandpass Signal Spectrum Ø A time portion of a bandpass signal. Notice the carrier and the baseband envelope. Time Waveform of Bandpass Signal 2

DEFINITIONS The Bandpass communication signal is obtained by modulating a baseband analog or digital signal onto a carrier. Definitions: Ø A baseband waveform has a spectral magnitude that is nonzero for frequencies in the vicinity of the origin ( f=0) and negligible elsewhere. Ø A bandpass waveform has a spectral magnitude that is nonzero for frequencies in some band concentrated about a frequency where fc>>0. fc-Carrier frequency Ø Modulation is process of imparting the source information onto a bandpass signal with a carrier frequency fc by the introduction of amplitude or phase perturbations or both. Ø This bandpass signal is called the modulated signal s(t), and the baseband source signal is called the modulating signal m(t). Information Signal input processing m Carrier circuits Transmission medium (channel) Communication System Carrier circuits Signal processing 3

Complex Envelope Representation Ø The waveforms g(t) , x(t), R(t), and are all baseband waveforms. Additionally all of them except g(t) are real and g(t) is the Complex Envelope. • g(t) is the Complex Envelope of v(t) • x(t) is said to be the In-phase modulation associated with v(t) • y(t) is said to be the Quadrature modulation associated with v(t) • R(t) is said to be the Amplitude modulation (AM) on v(t) • (t) is said to be the Phase modulation (PM) on v(t) In communications, frequencies in the baseband signal g(t) are said to be heterodyned up to fc Ø THEOREM: Any physical bandpass waveform v(t) can be represented as below where fc is the CARRIER frequency and c=2 fc 4

Generalized transmitter using the AM–PM generation technique. 5

Generalized transmitter using the quadrature generation technique. 6

Complex Envelope Representation Ø THEOREM: Any physical bandpass waveform v(t) can be represented by where fc is the CARRIER frequency and c=2 fc PROOF: Any physical waveform may be represented by the Complex Fourier Series The physical waveform is real, and using , Thus we have: cn - negligible magnitudes for n in the vicinity of 0 and, in particular, c 0=0 Introducing an arbitrary parameter fc , we get v(t) – bandpass waveform with non-zero spectrum concentrated near f=fc => cn – non-zero for ‘n’ in the range => g(t) – has a spectrum concentrated near f=0 (i. e. , g(t) - baseband waveform) 7

Complex Envelope Representation Ø Equivalent representations of the Bandpass signals: Ø Converting from one form to the other form Inphase and Quadrature (IQ) Components. Envelope and Phase Components 8

Complex Envelope Representation Ø The complex envelope resulting from x(t) being a computer generated voice signal and y(t) being a sinusoid. The spectrum of the bandpass signal generated from above signal. 9

Representation of Modulated Signals Ø Modulation is the process of encoding the source information m(t) into a bandpass signal s(t). Modulated signal is just a special application of the bandpass representation. The modulated signal is given by: • The complex envelope g(t) is a function of the modulating signal m(t) and is given by: g(t)=g[m(t)] where g[ • ] performs a mapping operation on m(t). • The g[m] functions that are easy to implement and that will give desirable spectral properties for different modulations are given by the TABLE 4. 1 • At receiver the inverse function m[g] will be implemented to recover the message. Mapping should suppress as much noise as possible during the recovery. 10

Bandpass Signal Conversion Ø On off Keying (Amplitude Modulation) of a unipolar line coded signal for bandpass conversion. 1 Xn 0 Unipolar Line Coder 1 g(t) 0 1 X cos( ct) 11

Bandpass Signal Conversion Ø Binary Phase Shift keying (Phase Modulation) of a polar line code for bandpass conversion. 1 Xn 0 Polar Line Coder 1 g(t) 0 1 X cos( ct) 12

Mapping Functions for Various Modulations 13

Envelope and Phase for Various Modulations 14

Spectrum of Bandpass Signals Theorem: If bandpass waveform is represented by Where is PSD of g(t) Proof: Thus, Using and the frequency translation property: We get, 15

PSD of Bandpass Signals Ø PSD is obtained by first evaluating the autocorrelation for v(t): Using the identity where and We get - Linear operators => or but AC reduces to PSD => 16

Evaluation of Power Theorem: Total average normalized power of a bandpass waveform v(t) is Proof: But So, or But is always real So, 17

Example : Amplitude-Modulated Signal Ø Evaluate the magnitude spectrum for an AM signal: Complex envelope of an AM signal: Spectrum of the complex envelope: AM signal waveform: AM spectrum: Magnitude spectrum: 18

Example : Amplitude-Modulated Signal Spectrum of AM signal. 19

Example : Amplitude-Modulated Signal ØTotal average power: 20

Study Examples SA 4 -1. Voltage spectrum of an AM signal Properties of the AM signal are: g(t)=Ac[1+m(t)]; Ac=500 V; m(t)=0. 8 sin(2 1000 t); fc=1150 k. Hz; Fourier transform of m(t): Spectrum of AM signal: Substituting the values of Ac and M(f), we have EEE 360 21

Study Examples SA 4 -2. PSD for an AM signal Autocorrelation for a sinusoidal signal (A sin w 0 t ) Autocorrelation for the complex envelope of the AM signal is Thus Using PSD for an AM signal: EEE 360 22

Study Examples SA 4 -3. Average power for an AM signal Normalized average power Alternate method: area under PDF for s(t) Actual average power dissipated in the 50 ohm load: SA 4 -4. PEP for an AM signal Normalized PEP: Actual PEP for this AM voltage signal with a 50 ohm load: 23