Chapter 2. SignalsHusheng LiThe University of Tennessee

Homework 2Deadline: 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 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 bandwidth

3 dB bandwidth

Noise equivalent bandwidth

Null-to-null bandwidth

Occupied bandwidth

Relative power spectrum bandwidth