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 University of Tennessee
Deadline: Sept. 16, 2013
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:
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.
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.
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.
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:
Each LTI system can be represented by its transfer function.
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.
High frequency attenuated
Low frequency attenuated
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.
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
Digital transversal filter
Linea distortion is theoretically curable through the use of equalization networks.
The multiple paths in wireless communications cause different delays along different paths, thus causing inter-symbol interference.
For example, consider two paths:
Many devices could have nonlinear transfer characteristics.
The nonlinear transfer characteristic may arouse harmonics.
Power gain: g=P_out / P_in
dB scale: g_dB = 10 log_10 g
For linear system of communication channel, we have
For the case of free-space transmission, the loss is given by
Consider the antenna gains, the received power is given by
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
Deadline: Sept. 9, 2013
An ideal bandpassfilter is given by
Perfect bandlimitiing and timelimiting are mutually incompatible.
Rise time is a measure of the ‘speed’ of a step response:
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.
A bandpass signal has the following frequency domain property:
The time domain bandpass signal can be written as
A bandpass signal can be decomposed to in-phase and quadrature components:
The frequency domain of a bandpass signal is given by
The in-phase and quadrature functions must be lowpass signals:
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
The connection between and is given by
In the frequency domain, we have
We can work on the lowpass equivalent spectra directly:
If the phase shift is nonlinear, we can approximate it by using the Taylor’s expansion:
A large bandwidth requires high carrier frequency.
3 dB bandwidth
Noise equivalent bandwidth
Relative power spectrum bandwidth