Signals in the time and frequency domains (lý THUYẾT TÍNH HIỆU SLIDE)

• An analog signal varies continuously with time, and has an infinite number of possible signal levels.. • A digital signal is discrete, but each sample has a finite number of possible s

3 Fourier Series Expansion of Periodic Signals

4 Spectral Representation of Periodic Signals

5 Duty Cycle of a Rectangular Wave

6 RMS Voltage and Power Spectrum of Aperiodic Signals

7 Discrete Fourier Transform (DFT)

8 Signal Strength

9 Signal Bandwidth

10 Conclusion

Introduction

• The tasks of communication is to encode information as a signal level, transmit this signal, then decode the signal at the receiving end

• An analog signal varies continuously with time, and has an infinite number of possible signal levels

• A discrete signal changes only once during a certain time interval The signal value during this time interval is one sample, and the interval length is called the sampling period Each sample has a infinite number of possible signal levels

• A digital signal is discrete, but each sample has a finite number

of possible signal levels The limited number of levels means that each sample transmits a single information It also means that each sample can be represented as digital data, a string of ones and zeroes

• A digital signal is preferred in computer communications because computers already store and process information digitally

• A digital signal will still be subject to noise, but the difference between signal levels (an

“0101” and an “0110”) will be sufficiently large

so that the receiver can always determine the original signal level in each sampling period The regenerated, so that the received digital data is an exact replica of the original digital data

• Analog to digital conversion reduces the amount of information of the signal by

approximating the analog signal with a digital signal The sampling period and number of levels

of the digital signal should be selected in order to capture as much information of the original signal

as possible

signal with phase = 0

signal with phase =/4

=/4

1 We assume that outside of the sampling interval, the signal repeats itself, so that the signal is periodic.

2 An alternative assumption is that the time-domain signal has a zero value outside of the sampling interval, so that the signal is aperiodic

• A periodic signal satisfies the condition:

• “Period” of the signal

Fourier Series Expansion of Periodic

Signals

a There are a finite number of discontinuities in the period T

b It has a finite average value for the period T.

c It has a finite number of positive and negative maxima in the

2 ( cos )

(

n

n n

a t

frequency,

n · f 0 : Frequency of each term

a n , b n : Fourier series coefficients :

t d t

S T

t d t n f t

S T

( cos ) (

2



dt t n f t

S T

0

) 2

( sin ) (

t

d n

t

d t

d

t d

A d

A

0

0 0

0

cos sin

3

cos 3

3 sin 2

cos 2

2 sin

cos sin

4 1

t t

t t

t A

0

0 0

0 0

0

cos 1

9

cos 9

1 7

cos 7 1

5

cos 5

1 3

cos 3

1 cos

n

t

n n

t t

t t

A A

0 2

1 2

0 0

0 0

cos 1

2 )

1 (

6

cos 35

2 4

cos 15 2

2

cos 3

2 cos

a t

2 ( cos )

(

n

n n

n n f t b n f t a

t

all bn=0

S(t)

t

Duty Cycle of a Rectangular Wave

period

mark of

T

τ T

d n

t

d t

d

t d

A d

A

0

0 0

0

cos sin

3

cos 3

3

sin 2

cos 2

2 sin

cos sin

4 1

cos 9

587 0 8

cos 8

951 0

7

cos 7

951 0 6

cos 6

587 0

0 4

cos 4

587 0 3

cos 3

951 0

2

cos 2

951 0 cos

587 0

4 6

0 )

(

0 0

0 0

0 0

0 0

% 20

t t

t t

t t

t t

A A

T

T 5

RMS Voltage Values and Power Spectrum

• The RMS voltage measures the signal’s power; it is the square root of the average value of the voltage squared, taken over one period of the signal:

 S t  d t T

V

T RMS  

0

2

)(1

For a sinusoidal signal :

T

t d t

S T

0

2 2

0

2

) cos(

1 )

)22

(cos1

1

0

m

T m

RMS

V dt

t T

1 because

0 0

T

T

t T

X( f ) has continuous variation in both the amplitude spectrum and

the phase spectrum Because each point in the amplitude spectrum

of an Aperiodic signal is infinitesimally higher in frequency than the last point, the voltage at any point in the amplitude spectrum is infinitesimally small

The output of the Fourier transform is therefore not voltage, but the

power spectral density of the waveform This value is the voltage

of a single spectral line whose power equals the amount of power contained an a 1 Hz wide frequency band with constant spectral density

t

A

τ 2 τ

S(t)

τ 2

A

f

f A

Phase = 0 for all f

• Match the fundamental frequency of the signal to the sampling interval If the sampling interval contains an integer number of signal periods, the harmonics of the periodic signal will appear as sharp peaks on the amplitude spectrum.

A rectangular wave with a frequency of 2000 Hz must be sampled with a Digital Spectrum Analyzer that samples at a rate of 500,000 samples per second How many samples should be taken if the sampling interval must contain 15 complete waveforms?

erval sampling

semples

int

3750 second

waveforms 2000

second

samples 000

, 500 interval

sampling waveforms

Signal Strength

• In signal analysis we frequently want to compare the power of two signals

• The decibel (dB) provides a convenient way to measure the difference

of two power levels

• It measures the logarithmic power difference between two signals

• If a signal is attenuated by 2 dB in one stage of transmission, then is attenuated by 5 dB in the next stage of transmission, we the two

decibel measurements to find the total attenuation of the two stages,

in this case 7 dB.

• Signal power equals to waveform power – DC power

2

1 10

log

10

P P

NdB  

Signal Bandwidth

• A baseband signal is any signal which transmits data in the form of the amplitude of the signal voltage The bandwidth of a baseband signal is

measured from zero frequency upwards to f max  

• A modulated signal transmits data by modifying the amplitude, frequency, or phase of a carrier signal The bandwidth of a modulated signal is measured from the minimum frequency f min   (below the carrier frequency) upwards to f max   (above the carrier frequency)

• The full bandwidth of a signal is the frequency range that includes all spectrum lines of the signal

• The absolute bandwidth (ABW) of a signal is the width of the spectrum that contains 98% of the signal’s total power

• The effective bandwidth (EBW) (bandwidth) of a signal is the width

of the spectrum that contains at least 50% of the signal’s total power This is the part of the signal whose power is within 3 dB of the complete signal.

C. The amplitude spectrum and phase spectrum together allow us to reconstruct the signal in the time domain If only the amplitude spectrum or power spectrum is available, it is possible to make some conclusions about features of the signal in the time domain.

D. The presence of a DC offset means that the signal has a constant component, and the

entire signal is shifted along the voltage axis in the time domain. 

I. If all harmonics divisible by a number n are missing for a rectangular periodic signal, the rectangular signal in the time domain has a duty cycle d equal to 1 /n .

F. The power spectrum allows us to determine the full bandwidth and the effective bandwidth of the signal.

G. Bandwidth refers to the range of frequencies represented in an analog signal The bandwidth of an analog signal determines the maximum sampling rate for a digital signal that is accurately transmitted via this analog signal.

Error Analysis

• A The absolute error of a measurement is the difference between

the ideal value x theory of a quantity (the measurement predicted by

theory) and the experimentally obtained value x measured

t

x x

For parts of the experiment where several measurements are taken at

once (a power spectrum measurement on the spectrum analyzer), you

should compare the error of each measurement against the maximum

theoretical value in that set of measurements (generally the theoretical amplitude of the first harmonic)

maximum

) ( theory

t measuremen