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. Time Waveform of Bandpass Signal
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
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 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).
Transmission
medium
(channel)
Carrier
circuits
Signal
processing
Information m
input
Communication System

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

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

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

14. Mapping Functions for Various Modulations

15. 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,

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
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:

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

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

22. 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(21000t); fc=1150 kHz;
Fourier transform of m(t):
Spectrum of AM signal:
Substituting the values of Ac and M(f), we have

23. Study Examples 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
Thus
Using
PSD for an AM signal:

24. 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: