1. Chapter4 Bandpass Signalling Bandpass Filtering and Linear Distortion
Bandpass Sampling Theorem
Bandpass Dimensionality Theorem
Amplifiers and Nonlinear Distortion
Total Harmonic Distortion (THD)
Intermodulation Distortion (IMD)
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University

2. Bandpass Filtering and Linear Distortion
Equivalent Low-pass filter: Modeling a bandpass filter by using an equivalent low pass filter (complex impulse response)
Bandpass filter
H(f) = Y(f)/X(f)
Input bandpass waveform
Output bandpass waveform
Impulse response of the bandpass filter
Frequency response of the bandpass filter

3. Bandpass Filtering

4. Bandpass Filtering Theorem:
The complex envelopes for the input, output, and impulse response of a bandpass filter are related by
g1(t) – complex envelope of input
k(t) – complex envelope of impulse response
Also,
Proof:
Spectrum of the output is
Spectra of bandpass waveforms are related to that of their complex enveloped
But

5. Bandpass Filtering Thus, we see that
Taking inverse fourier transform on both sides
Any bandpass filter may be described and analyzed by using an equivalent low-pass filter.
Equations for equivalent LPF are usually much less complicated than those for bandpass filters & so the equivalent LPF system model is very useful.

6. Linear Distortion
For distortionless transmission of bandpass signals, the channel transfer function H(f) should satisfy the following requirements:
The amplitude response is constant
A- positive constant
The derivative of the phase response is constant
Tg – complex envelope delay
Integrating the above equation, we get
Are these requirements sufficient for distortionless transmission?

7. Linear Distortion

8. Bandpass filter delays input info by Tg , whereas the carrier by Td
Linear Distortion
The channel transfer function is
If the input to the bandpass channel is
Then the output to the channel (considering the delay Tg due to ) is
Using
Bandpass filter delays input info by Tg , whereas the carrier by Td
Modulation on the carrier is delayed by Tg & carrier by Td

9. Bandpass Sampling Theorem
If a waveform has a non-zero spectrum only over the interval , where the transmission bandwidth BT is taken to be same as absolute BW, BT=f2-f1, then the waveform may be reproduced by its sample values if the sampling rate is
Quadrature bandpass representation
Let fc be center of the bandpass:
x(t) and y(t) are absolutely bandlimited to B=BT/2
The sampling rate required to represent the baseband signal is
Quadrature bandpass representation now becomes
Where and samples are independent , two sample values are obtained for each value of n
Overall sampling rate for v(t):

10. Bandpass Dimensionality Theorem
Assume that a bandpass waveform has a nonzero spectrum only over a frequency interval , where the transmission bandwidth BT is taken to be the absolute bandwidth given by BT=f2-f1 and BT<<f1.
The waveform may be completely specified over a T0-second interval by N Independent pieces of information. N is said to be the number of dimensions required to specify the information.

11. Received Signal Pulse The signal out of the transmitter
Transmission
medium
(Channel)
Carrier
circuits
Signal
processing
Information m
input
The signal out of the transmitter
g(t) – Complex envelope of v(t)
If the channel is LTI , then received signal + noise
n(t) – Noise at the receiver input
Signal + noise at the receiver input
A – gain of the channel
- carrier phase shift caused by the channel, Tg – channel group delay.
Signal + noise at the receiver input

12. Nonlinear Distortion Amplifiers Non-linear Linear
Circuits with memory and circuits with no memory
Memory - Present output value ~ function of present input + previous input values
- contain L & C
No memory - Present output values ~ function only of its present input values.
Circuits : linear + no memory – resistive ciruits
- linear + memory – RLC ciruits (Transfer function)

13. Nonlinear Distortion
Assume no memory  Present output as a function of present input in ‘t’ domain
If the amplifier is linear
K- voltage gain of the amplifier
In practice, amplifier output becomes saturated as the amplitude of the input signal is increased.
output-to-input characteristic (Taylor’s expansion):
Where
- output dc offset level
- 1st order (linear) term
- 2nd order (square law) term

14. 2nd Harmonic Distortion with
Nonlinear Distortion
Harmonic Distortion associated with the amplifier output:
Let the input test tone be represented by
To the amplifier input
Then the second-order output term is
2nd Harmonic Distortion with =
In general, for a single-tone input, the output will be
Vn – peak value of the output at the frequency nf0
The Percentage Total Harmonic Distortion (THD) of an amplifier is defined by

15. Nonlinear Distortion
Intermodulation distortion (IMD) of the amplifier:
If the input (tone) signals are
Then the second-order output term is
IMD
Harmonic distortion at 2f1 & 2f2
Second-order IMD is:

16. Main Distortion Products
Nonlinear Distortion
Third order term is
The second term (cross-product) is
The third term is
Intermodulation terms at nonharmonic frequencies
For bandpass amplifiers, where f1 & f2 are within the pasband, f1 close to f2, the distortion products at 2f1+f2 and 2f2+f1 ~ outside the passband
Main Distortion Products