Properties of Signals

Prof. Satheesh Monikandan.BHOD-ECE

INDIAN NAVAL ACADEMY, EZHIMALA

sathy24@gmail.com

92 INAC-L-AT15

Comparison of analog and digital signals

Signals can be analog or digital. Analog signals can have an infinite number of values in a range; digital signals can have only a limited number of values.

Phase

The term phase describes the position of the waveform relative to time zero.

The phase is measured in degrees or radians (360 degrees is 2π radians)

Signal Energy and Power

Total energy of a continuous signal x(t) over [t1, t2] is:

where |.| denote the magnitude of the (complex) number.

Similarly for a discrete time signal x[n] over [n1, n2]:

By dividing the quantities by (t2-t1) and (n2-n1+1), respectively, gives the average power, P

Note that these are similar to the electrical analogies (voltage), but they are different, both value and dimension.

E=∫t1

t2∣x ( t )∣2 dt

E=∑n=n1

n2∣x [ n ]∣2

Energy and Power over Infinite TimeFor many signals, we’re interested in examining the power and energy

over an infinite time interval (-∞, ∞). These quantities are therefore defined by:

If the sums or integrals do not converge, the energy of such a signal is infinite.

Two important (sub)classes of signals

1. Finite total energy (and therefore zero average power)

2. Finite average power (and therefore infinite total energy)

E∞=limT→∞∫−T

T∣x ( t )∣2dt=∫−∞

∞∣x ( t )∣2 dt

E∞=limN →∞∑n=−N

N∣x [ n ]∣

2=∑n=−∞

∞∣x [ n ]∣2

P∞=limT →∞

12T∫−T

T∣x ( t )∣2 dt

P∞=limN →∞

12N+1∑n=−N

N∣x [ n ]∣2

An important class of signals is the class of periodic signals. A periodic signal is a continuous time signal x(t), that has the property

where T>0, for all t.

Examples:cos(t+2π) = cos(t)sin(t+2π) = sin(t)Both are periodic with period 2π

For a signal to be periodic, the relationship must hold for all t.

Periodic Signals

x ( t )=x ( t+T )2π

An even signal is identical to its time reversed signal, i.e. it can be reflected in the origin and is equal to the original:

Examples:x(t) = cos(t)

An odd signal is identical to its negated, time reversed signal, i.e. it is equal to the negative reflected signal

Examples:x(t) = sin(t)x(t) = t

This is important because any signal can be expressed as the sum of an odd signal and an even signal.

Odd and Even Signals

x (−t )=x ( t )

x (−t )=−x ( t )

Exponential and Sinusoidal Signals

Exponential and sinusoidal signals are characteristic of real-world signals and also from a basis (a building block) for other signals.

A generic complex exponential signal is of the form:

where C and a are, in general, complex numbers.

Real exponential signals

x ( t )=Ceat

a>0C>0

a<0C>0

Exponential growth Exponential decay

Periodic Complex Exponential & Sinusoidal Signals

Consider when a is purely imaginary:

By Euler’s relationship, this can be expressed as:

This is a periodic signals because:

when T=2π/ω0

A closely related signal is the sinusoidal signal:

We can always use:

x ( t )=Cejω0t

ejω

0t=cosω0t+ j sinω0 t

ejω0( t+T )

=cosω0( t+T )+ j sinω0( t+T )

=cosω0 t+ j sinω0 t=ejω0 t

x ( t )=cos (ω0 t+φ ) ω0=2πf 0

A cos (ω0 t+φ )=Aℜ (e j (ω0 t +φ ))

A sin (ω0 t+φ )=Aℑ (e j(ω0 t +φ ))

T0 = 2π/ω0

cos(1)

T0 is the fundamental time periodω0 is the fundamental frequency

Exponential & Sinusoidal Signal PropertiesPeriodic signals, in particular complex periodic

and sinusoidal signals, have infinite total energy but finite average power.

Consider energy over one period:

Therefore:

Average power:

Useful to consider harmonic signals

Terminology is consistent with its use in music, where each frequency is an integer multiple of a fundamental frequency.

E period=∫0

T 0∣ejω0 t∣

2dt

=∫0

T0 1 dt=T 0

P period=1T 0

E period=1

E∞=∞

General Complex Exponential Signals

So far, considered the real and periodic complex exponential

Now consider when C can be complex. Let us express C is polar form and a in rectangular form:

So

Using Euler’s relation

These are damped sinusoids

C=∣C∣e jφ

a=r+ jω0

Ceat=∣C∣e jφ e( r+ jω0 )t

=∣C∣e rt ej(ω

0+φ)t

Ceat=∣C∣e jφ e( r+ jω0 )t

=∣C∣e rt cos( (ω0+φ) t )+ j∣C∣e rtsin ( (ω0 +φ ) t )

Discrete Unit Impulse and Step Signals

The discrete unit impulse signal is defined:

Useful as a basis for analyzing other signals

The discrete unit step signal is defined:

Note that the unit impulse is the first difference (derivative) of the step signal

Similarly, the unit step is the running sum (integral) of the unit impulse.

x [ n ]=δ [ n ]={0 n≠01 n=0

x [ n ]=u[ n ]={0 n<01 n≥0

δ [ n ]=u [ n ]−u [ n−1 ]

Continuous Unit Impulse and Step Signals

The continuous unit impulse signal is defined:

Note that it is discontinuous at t=0

The arrow is used to denote area, rather than actual value

Again, useful for an infinite basis

The continuous unit step signal is defined:

x ( t )=δ ( t )={0 t≠0∞ t=0

x ( t )=u ( t )=∫−∞

tδ (τ )dτ

x ( t )=u ( t )={0 t<01 t>0

TIME AND FREQUENCY DOMAINS

•Time-domain representation

•Frequency-domain representation

Time and frequency domains

Periodic Composite SignalsA single-frequency sine wave is not useful in data communications; we need to change one or more of its characteristics to make it useful.

According to Fourier analysis, any composite signal can be represented as a combination of simple sine waves with different frequencies, phases, and amplitudes.

Frequency spectrum comparison

Frequency Spectrum and Bandwidth

The frequency spectrum of a signal is the collection of all the component frequencies it contains and is shown using a frequency-domain graph.

The bandwidth of a signal is the width of the frequency spectrum, i.e., bandwidth refers to the range of component frequencies.

To compute the bandwidth, subtract the lowest frequency from the highest frequency of the range.

Example 1Example 1

If a periodic signal is decomposed into five sine waves with frequencies of 100, 300, 500, 700, and 900 Hz, what is the bandwidth? Draw the spectrum, assuming all components have a maximum amplitude of 10 V.

SolutionSolution

B = fh − fl = 900 − 100 = 800 HzThe spectrum has only five spikes, at 100, 300, 500, 700, and 900 (see Figure 13.4 )

Example 1