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 in- phase 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 and 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