Chapter4 Bandpass Signalling Definitions Complex Envelope Representation

1. Chapter4 • Bandpass Signalling • Definitions • Complex Envelope Representation • Representation of Modulated Signals • Spectrum of Bandpass Signals • Power of Bandpass Signals • Examples Huseyin Bilgekul Eeng360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University

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

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

4. Complex Envelope Representation • 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 • THEOREM: Any physical bandpass waveformv(t) can be represented as below where fc is the CARRIER frequency and c=2 fc

5. Generalized transmitter using the AM–PM generation technique.

6. Generalized transmitter using the quadrature generation technique.

7. Complex Envelope Representation • THEOREM: Any physical bandpass waveformv(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, c0=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)

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

9. Complex Envelope Representation • 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.

10. Representation of Modulated Signals • 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.

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

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

14. Mapping Functions for Various Modulations

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

16. PSD of Bandpass Signals • 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 =>

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

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

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

21. Example : Amplitude-Modulated Signal • Total average normalized power:

22. Study Examples Fourier transform of m(t): Spectrum of AM signal: SA4-1.Voltage spectrum of an AM signal Properties of the AM signal are: g(t)=Ac[1+m(t)]; Ac=500 V; m(t)=0.8sin(21000t); fc=1150 kHz; Substituting the values of Ac and M(f), we have

23. Study Examples Thus Using PSD for an AM signal: SA4-2. PSD for an AM signal Autocorrelation for a sinusoidal signal (A sin w0t ) Autocorrelation for the complex envelope of the AM signal is

24. Study Examples Normalized PEP: SA4-3. 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: SA4-4. PEP for an AM signal Actual PEP for this AM voltage signal with a 50 ohm load: