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