- 1. Eeng 360 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. Eeng 360 2 Energy spectrum of a bandpass signal is concentrated around the carrier frequency fc. A time portion of a bandpass signal. Notice the carrier and the baseband envelope. Bandpass Signals Bandpass Signal Spectrum Time Waveform of Bandpass Signal
- 3. Eeng 360 3 DEFINITIONS Definitions: The Bandpass communication signal is obtained by modulating a baseband analog or digital signal onto a carrier. A baseband waveform has a spectral magnitude that is nonzero for frequencies in the vicinity of the origin ( f=0) and negligible elsewhere. 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 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). c f f Transmission medium (channel) Carrier circuits Signal processing Carrier circuits Signal processing Information m input m ~ ) ( ~ t g ) (t r ) (t s ) (t g Communication System
- 4. Eeng 360 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. t • 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 waveform v(t) can be represented as below where fc is the CARRIER frequency and c=2 fc ( ) ( ) ( ) ( ) ( ) ( ) j t j g t g t x t jy t g t e R t e Re cos = cos sin c j t c c c v t g t e R t t t x t t y t t
- 5. Eeng 360 5 Generalized transmitter using the AM–PM generation technique.
- 6. Eeng 360 6 Generalized transmitter using the quadrature generation technique.
- 7. Eeng 360 7 v(t) – bandpass waveform with non-zero spectrum concentrated near f=fc => cn – non-zero for ‘n’ in the range The physical waveform is real, and using , Thus we have: Complex Envelope Representation 0 0 0 ( ) 2 / n jn t n n v t c e T 0 0 1 Re 2 jn t n n v t c c e 0 1 Re ( ) Re 2 c c c n j n t j t j t n n v t g t e c e e 0 ( ) 1 ( ) 2 c j n t n n g t c e 0 c nf f PROOF: Any physical waveform may be represented by the Complex Fourier Series * n n c c cn - negligible magnitudes for n in the vicinity of 0 and, in particular, c0=0 Introducing an arbitrary parameter fc , we get => g(t) – has a spectrum concentrated near f=0 (i.e., g(t) - baseband waveform) * 1 1 Re 2 2 THEOREM: Any physical bandpass waveform v(t) can be represented by where fc is the CARRIER frequency and c=2 fc Re c j t v t g t e
- 8. Eeng 360 8 Converting from one form to the other form Equivalent representations of the Bandpass signals: Complex Envelope Representation cos sin Inphase and Quadrature (IQ) form c c v t x t t y t t ( ) ( ) ( ) ( ) Complex Envelope of ( ) j g t j t g t x t jy t g t e R t e v t Inphase and Quadrature (IQ) Components. Re ( )cos ( ) Im ( )sin ( ) x t g t R t t y t g t R t t Envelope and Phase Components 2 2 1 ( ) ( ) ( ) ( ) ( ) ( ) tan ( ) ( ) R t g t x t y t y t t g t x t Re cos Envelope and Phase form c j t c v t g t e R t t t
- 9. Eeng 360 9 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. Complex Envelope Representation
- 10. Eeng 360 10 Representation of Modulated Signals • 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. 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: Re ( ) 2 c j t c c s t g t e f
- 11. Eeng 360 11 Bandpass Signal Conversion ) (t g ) (t s 1 1 1 0 0 2 Ac 2 0 Ac 2 n X X Unipolar Line Coder cos(ct) g(t) Xn c A ) (t s On off Keying (Amplitude Modulation) of a unipolar line coded signal for bandpass conversion.
- 12. Eeng 360 12 Binary Phase Shift keying (Phase Modulation) of a polar line code for bandpass conversion. X Polar Line Coder cos(ct) g(t) Xn c A ) (t s ) (t g ) (t s 1 1 1 0 0 2 Ac 2 2 Ac 2 n X Bandpass Signal Conversion
- 13. Eeng 360 13 Mapping Functions for Various Modulations
- 14. Eeng 360 14 Envelope and Phase for Various Modulations
- 15. Eeng 360 15 Spectrum of Bandpass Signals * 1 1 Re ( ) ( ) 2 2 c c c j t j t j t v t g t e g t e g t e t j t j c c e t g F e t g F t v F f V * 2 1 2 1 ) ( f G t g F * * * 1 ( ) - - 2 c c V f G f f G f f Theorem: If bandpass waveform is represented by t g F f G f Pg Where is PSD of g(t) Proof: Thus, Using and the frequency translation property: We get, * 1 Spectrum of Bandpass Signal ( ) 2 1 PSD of Bandpass Signal ( ) 4 c c v g c g c V f G f f G f f P f P f f P f f Re ( ) c j t v t g t e
- 16. Eeng 360 16 PSD of Bandpass Signals Re Re c c j t j t v R v t v t g t e g t e * 2 1 2 1 2 1 1 1 Re Re Re Re 2 2 c c c c c c 2 ( ) c j t c g t e 1 c j t c g t e * 1 1 Re Re 2 2 c c c c j t j t j t j t v R g t g t e e g t g t e e , 2 * 1 1 Re Re 2 2 c c c j j t j v R g t g t e g t g t e e 2 * 1 1 Re Re 2 2 c c c j j t j v R g t g t e g t g t e e PSD is obtained by first evaluating the autocorrelation for v(t): Using the identity where and - Linear operators * g g t g t R but 1 Re 2 c j v g R R e AC reduces to * * 1 ( ) 4 v v g c g c g g P f F R P f f P f f P P f PSD => => We get or g(t) fc in s frequencie
- 17. Eeng 360 17 Evaluation of Power 2 v v P v t P f df 1 2 j f v v v R F P f P f e df 0 v v R P f df * 1 1 1 0 Re 0 Re 0 Since Re 2 2 2 c j v g v g R R g t g t R R e 2 1 0 Re 2 v R g t 2 1 0 2 v R g t g t Theorem: Total average normalized power of a bandpass waveform v(t) is Proof: But So, or But is always real So, 2 2 1 0 2 v v v P v t P f df R g t
- 18. Eeng 360 18 Example : Amplitude-Modulated Signal Evaluate the magnitude spectrum for an AM signal: 1 c g t A m t Complex envelope of an AM signal: c c G f A f A M f Spectrum of the complex envelope: 1 2 c c c c c S f A f f M f f f f M f f AM spectrum: 1 1 , 0 2 2 1 1 , 0 2 2 A f f A M f f f c c c c S f A f f A M f f f c c c c Magnitude spectrum: AM signal waveform: Re ( ) 1 cos c j t c c s t g t e A m t t * * * Because ( ) is real and do not over 1 ( ) - - 2 and lap c c c c S f G f f G f f M f M f f f G f f G f f m t
- 19. Eeng 360 19 Example : Amplitude-Modulated Signal Spectrum of AM signal.
- 20. Eeng 360 20 2 2 2 2 2 2 2 2 Side 2 b 2 a 2 nd 2 2 If DC value of ( ) is zero 1 2 Carrier 1 Powe 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 1 2 Where 1 2 Sideband Po er r w s c c c c c c m c m m c P g t A m t A m t m t A m t m t A m t A A P P P m t A P t P m Example : Amplitude-Modulated Signal Total average power:
- 21. EEE 360 21 Study Examples 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(21000t); fc=1150 kHz; 2 1000 2 1000 0.8 0.8sin 2 1000 2 j t j t m t t e e j 250 100 1000 100 1000 250 100 1000 100 1000 c c c c c c S f f f j f f j f f f f j f f j f f 0.4 1000 0.4 1000 M f j f j f Fourier transform of m(t): 1 2 c c c c c S f A f f M f f f f M f f Spectrum of AM signal: Substituting the values of Ac and M(f), we have
- 22. EEE 360 22 0 0 2 2 0 o cos A=0.8 and 2 1000 2 2 j j m A A R e e 2 2 0 0 1000 1000 4 4 m A A P f f f f f f f 2 2 * 1 1 1 g c c R g t g t A m t m t A m t m t m t m t 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 Study Examples 2 But 1 1, 0, , 1 m g c m m t m t m t m t R R A R 2 2 g c c m P f A f A P f 2 1 g c m R A R Thus 1 ( ) 4 v g c g c P f P f f P f f Using 62500 10000 1000 10000 1000 62500 10000 1000 10000 1000 s c c c c c c P f f f f f f f f f f f f f PSD for an AM signal:
- 23. EEE 360 23 Study Examples 2 165 s s s norm rms P V P f df kW 2 5 1.65 10 3.3 kW 50 s rms s norm L V P R 2 2 2 2 2 1 1 0.8 1 500 1 165 2 2 2 s s c m norm rms rms P V A V kW 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: 2 5 4.05 10 8.1 50 PEP rms PEP actual L P P kW R SA4-4. PEP for an AM signal 2 2 2 2 2 1 1 1 max 1 max 500 1 0.8 405 2 2 2 PEP c norm P g t A m t kW Normalized PEP: Actual PEP for this AM voltage signal with a 50 ohm load: