Transcript Chapter2
Chapter 2. Signals
Husheng Li
The University of Tennessee
Homework 2
Deadline: Sept. 16, 2013
Spectrum
Physically, the signal is transmitted in the time
domain.
It is more convenient to study the signal in the
frequency domain.
The frequency domain description is called the
spectrum.
The frequency description of signal can be
obtained from Fourier transform:
Example: Rectangular Pulse
Time domain
Frequency domain
Signal Energy
Rayleigh’s Theorem: The signal energy is
given by
Integrating the square of the amplitude
spectrum over all frequency yields the
total energy.
|V(f)|^2 is called the energy spectral
density.
Band Limited Signals
A signal should not use all bandwidth. Hence, we have to
limit its band.
Sinc function is a band limited one
A band limited signal is infinite in the time, which is
impossible in practice.
Frequency Translation
We need to transform a baseband signal to much
higher frequency one. (Why?)
It is equivalent to multiplying a sinusoidal signal having
the carrier frequency.
RF Pulse
time
frequency
Convolution
When a signal is passed through a linear time
invariant (LTI) system, the output is the
convolution of the input signal and the system
impulse response.
In the frequency domain, the convolution is
equivalent to multiplication:
Transfer Function
Each LTI system can be represented by its transfer function.
Signal Transmission:
Distortionless Case
The output is undistorted if it differs from the input
only by a multiplying constant and a finite time
delay:
In the frequency domain, it is equivalent to
In practice, the signal is always distorted.
Linear Distortion: Amplitude
Linear distortion includes any amplitude or
delay distortion associated with a linear
transmission system, which is easily descried
in the frequency domain.
The amplitude could be distorted.
Low frequency attenuated
High frequency attenuated
Linear Distortion: Phase
If the phase shift is not linear, the various
frequency components suffer different amounts
of time delay, called phase or delay distortion.
The delay is given by
Two Waveforms: Example
Equalization
Linea distortion is theoretically curable through
the use of equalization networks.
Digital transversal filter
Multipath in Wireless
The multiple paths in wireless communications
cause different delays along different paths, thus
causing inter-symbol interference.
For example, consider two paths:
Destructive Interference
(two-path)
Nonlinear Distortion
Many devices could have nonlinear transfer
characteristics.
The nonlinear transfer characteristic may arouse
harmonics.
Transmission Loss
Power gain: g=P_out / P_in
dB scale: g_dB = 10 log_10 g
For linear system of communication channel, we have
Typical Values of Power Loss
Example: Radio Transmission
For the case of free-space transmission, the loss is
given by
Consider the antenna gains, the received power
is given by
Example: Satellite
Communication
Doppler Shift
A passing automobile’s horn will appear to
change pitch as it passes by.
The change in frequency is called Doppler shift.
When the moving speed is v and the angle is ϕ,
the Dopper shift is
Homework
Deadline: Sept. 9, 2013
Ideal Filter
An ideal bandpass filter is given by
Filtering
Perfect bandlimitiing and timelimiting are mutually
incompatible.
Rise time is a measure of the ‘speed’ of a step
response:
Quadrature Filter
A quadrature filter is an allpass network that
merely shifts the phase of the positive frequency
components by -90 degrees.
The output of a quadrature filter is called the
Hilbert transform of the input.
Properties of Hilbert
Transform
Bandpass Signals and
Systems
A bandpass signal has the following frequency
domain property:
The time domain bandpass signal can be written
as
Spectrum and Waveform of
Bandpass Signal
Quadrature-Carrier
Description of Bandpass
Signal
A bandpass signal can be decomposed to inphase and quadrature components:
Frequency Domain of
Bandpass Signal
The frequency domain of a bandpass signal is
given by
The in-phase and quadrature functions must be
lowpass signals:
Lowpass Equivalent Signal
In the frequency domain, we have the low pass
equivalent spectrum:
In the time domain, we have the lowpass
equivalent signal:
In the frequency domain, we have
Lowpass-to-bandpass
transformation
The connection between vlp (t) andv bp (t) is given
by
In the frequency domain, we have
Bandpass Transmission
We can work on the lowpass equivalent spectra
directly:
Carrier and Envelop Delay
If the phase shift is nonlinear, we can
approximate it by using the Taylor’s expansion:
Bandwidth and Carrier
Frequency
A large bandwidth requires high carrier
frequency.
Bandwidth: Definition
Absolute bandwidth
3 dB bandwidth
Noise equivalent bandwidth
Null-to-null bandwidth
Occupied bandwidth
Relative power spectrum bandwidth