Eeng 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

Eeng  Energy spectrum of a bandpass signal is concentrated around the carrier frequency f c.  A time portion of a bandpass signal. Notice the carrier and the baseband envelope. Bandpass Signals Bandpass Signal Spectrum Time Waveform of Bandpass Signal

Eeng 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 f c >>0. f c -Carrier frequency  Modulation is process of imparting the source information onto a bandpass signal with a carrier frequency f c 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). Transmission medium (channel) Carrier circuits Signal processing Carrier circuits Signal processing Information m input Communication System

Eeng 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 f c  THEOREM: Any physical bandpass waveform v(t) can be represented as below where f c is the CARRIER frequency and  c =2  f c

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

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

Eeng  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

Eeng 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 m odulated signal is given by:

Eeng Bandpass Signal Conversion X Unipolar Line Coder cos(  c t) g(t)XnXn  On off Keying (Amplitude Modulation) of a unipolar line coded signal for bandpass conversion.

Eeng  Binary Phase Shift keying (Phase Modulation) of a polar line code for bandpass conversion. X Polar Line Coder cos(  c t) g(t)XnXn Bandpass Signal Conversion

Eeng Mapping Functions for Various Modulations

Eeng Envelope and Phase for Various Modulations

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

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

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

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

Eeng Example : Amplitude-Modulated Signal

Eeng Example : Amplitude-Modulated Signal  Total average power:

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

EEE SA4-2. PSD for an AM signal Autocorrelation for a sinusoidal signal (A sin w 0 t – ref ex. 2-10) A=0.8 and Autocorrelation for the complex envelope of the AM signal is Study Examples But Thus Using PSD for an AM signal:

EEE Study Examples 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 Normalized PEP: Actual PEP for this AM voltage signal with a 50 ohm load: