chapter 2. signals
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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. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 2. SignalsHusheng LiThe University of Tennessee
Homework 2Deadline: Sept. 16, 2013
SpectrumPhysically, 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 EnergyRayleigh’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
ConvolutionWhen 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
EqualizationLinea 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 TransmissionFor 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 ShiftA 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 FilterAn 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 FilterA 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 SystemsA 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 SignalA bandpass signal can be decomposed to in-
phase and quadrature components:
Frequency Domain of Bandpass SignalThe 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 transformationThe connection between and is
given by
In the frequency domain, we have
Bandpass TransmissionWe 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 FrequencyA large bandwidth requires high carrier
frequency.
Bandwidth: DefinitionAbsolute bandwidth3 dB bandwidthNoise equivalent bandwidthNull-to-null bandwidthOccupied bandwidthRelative power spectrum bandwidth