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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.

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chapter 2 signals

Chapter 2. Signals

Husheng Li

The University of Tennessee

homework 2
Homework 2

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:

example rectangular pulse
Example: Rectangular Pulse

Time domain

Frequency domain

signal energy
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
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
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
RF Pulse




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
Transfer Function

Each LTI system can be represented by its transfer function.

signal transmission distortionless case
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: Amplitude

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.

linear distortion phase
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


Digital transversal filter

Linea distortion is theoretically curable through the use of equalization networks.

multipath in wireless
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:

nonlinear distortion
Nonlinear Distortion

Many devices could have nonlinear transfer characteristics.

The nonlinear transfer characteristic may arouse harmonics.

transmission loss
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

example radio transmission
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

doppler shift
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


Deadline: Sept. 9, 2013

ideal filter
Ideal Filter

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:

quadrature filter
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.

bandpass signals and systems
Bandpass Signals and Systems

A bandpass signal has the following frequency domain property:

The time domain bandpass signal can be written as

quadrature carrier description of b andpass signal
Quadrature-Carrier Description of Bandpass Signal

A bandpass signal can be decomposed to in-phase and quadrature components:

frequency domain of bandpass signal
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
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
Lowpass-to-bandpass transformation

The connection between and is given by

In the frequency domain, we have

bandpass transmission
Bandpass Transmission

We can work on the lowpass equivalent spectra directly:

carrier and envelop delay
Carrier and Envelop Delay

If the phase shift is nonlinear, we can approximate it by using the Taylor’s expansion:

bandwidth and carrier frequency
Bandwidth and Carrier Frequency

A large bandwidth requires high carrier frequency.

bandwidth definition
Bandwidth: Definition

Absolute bandwidth

3 dB bandwidth

Noise equivalent bandwidth

Null-to-null bandwidth

Occupied bandwidth

Relative power spectrum bandwidth