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DISCRETE TIME SIGNALS

As has been said, the samples of a continuous-time signal form the corresponding discrete-time signal. These samples have to be quantized and binary-encoded to really become a digital signal. However the two last processes, quantization and coding, can be understood, thus when we say discrete-time and digital we usually mean the same thing, the two words are interchangeable.

Figure 1: Discrete-time signal

Figure 1 gives examples. The values of the samples x(n) can be anything: Positive or negative, zero or infinity, integer or fraction, real or complex (usually assumed real). The signal may be infinite duration, i.e. exist at all time, or finite duration, i.e exists for a short duration, usually taken around the origin.

One convenient way to describe discrete-time signals is to use sequence (vector). For example for the two signals in Figure 1 we write respectively

x ( n ) = [ . . . 1, − 2,2,3,1, − 1,2, − 2,1,3 . . . ]

x ( n ) = [ . . . 1, − 2,2,3,1, − 1,2, − 2,1,3 . . . ] size 12{x \( n \) = \[ "." "." "." 1, - 2,2,3,1, - 1,2, - 2,1,3 "." "." "." \] } {}

x ( n ) = [ − 2, − 1,2,2, − 1,3 ]

x ( n ) = [ − 2, − 1,2,2, − 1,3 ] size 12{x \( n \) = \[ - 2, - 1,2,2, - 1,3 \] } {}

By this reason, a discrete-time signal usually called a sequence. Notice that we have to specify the sample at origin, e.g. by writing it in bold face, or underlined, or with an arrow.

Basic discrete-time signals

In principle, discrete time signals are samples of corresponding continuous-time sources. Thus we also have basic discrete-time signals, similar to the continuous-time case (Section 1.12).

(a) Unit sample

Unit sample, also called unit impulse, is a signal having amplitude of 1 at origin, and zero otherwise ( Figure 2a):

δ ( n ) = 1 , n = 0 0, n ≠ 0

δ ( n ) = 1 , n = 0 0, n ≠ 0 alignl { stack { size 12{δ \( t \) = infinity , matrix { {} # {} # {} } t=0} {} # size 12{ matrix { {} # {} # {} } matrix { {} # {} } 0, matrix { {} # {} # {} } t <> 0} {} } } {}

(1)

Figure 2: Three simple basic signals

Notice that this discrete-time signal is not the sampled version of the analog counterpart (Section 1.1.2) but still δ(n)=δ(−n)

δ(n)=δ(−n) size 12{δ \( n \) =δ \( - n \) } {}

(Equation (1.4)).

(b) Unit step

The unit step is defined as ( Figure 2b)

u ( n ) = 1, n ≥ 0 0, n < 0 ( or n ≤ − 1 )

u ( n ) = 1, n ≥ 0 0, n < 0 ( or n ≤ − 1 ) alignl { stack { size 12{r \( n \) =0, matrix { {} # {} # {} } t >= 0} {} # size 12{ matrix { {} # {} # matrix { {} # {} } 1, matrix { {} # {} # {} } {} } t<0 \( ital "or" matrix { {} # {} } n <= - 1 \) } {} } } {}

(2)

(c) Unit ramp

This is a divergent signal (amplitude goes to ∞ as n goes to ∞), defined as (Figure 2c)

r ( n ) = n, n ≥ 0 0, n < 0 ( or n ≤ − 1 )

r ( n ) = n, n ≥ 0 0, n < 0 ( or n ≤ − 1 ) alignl { stack { size 12{r \( n \) =0, matrix { {} # {} # {} } t >= 0} {} # size 12{ matrix { {} # {} # matrix { {} # {} } 1, matrix { {} # {} # {} } {} } t<0 \( ital "or" matrix { {} # {} } n <= - 1 \) } {} } } {}

(3)

Figure 3: Real exponential

(d) Real exponential

The real exponential is quite a popular signal, defined as

x ( n ) = a n , n ≥ 0 0, n < 0

x ( n ) = a n , n ≥ 0 0, n < 0 alignl { stack { size 12{x \( n \) =a rSup { size 8{n} } , matrix { {} # {} # {} } n >= 0} {} # matrix { {} # {} # matrix { {} # {} } 0, matrix { {} # {} # {} } {} } n<0 {} } } {}

(4)

Where a is a real constant. There are four different cases as seen in Figure 3 in which two cases are convergent and two cases are divergent. The complex exponential with be discussed later.

1.4.2 Sinusoid, digital frequency, periodicity, complex exponential

The expression of a cosinusoidal signal (Equation 1.1) is sampled at period T s

T s size 12{x rSub { size 8{m} } \( ital "nT" \) } {}

x ( t ) = A cos ( Ω t + Φ 0 ) = A cos ( Ω nT s + Φ 0 )

x ( t ) = A cos ( Ω t + Φ 0 ) = A cos ( Ω nT s + Φ 0 ) size 12{x \( t \) =A"cos" \( %OMEGA t+Φ rSub { size 8{0} } \) =A"cos" \( %OMEGA ital "nT" rSub { size 8{s} } +Φ rSub { size 8{0} } \) } {}

(5)

where A is the amplitude, Ω = 2 ⁢ πF

Ω = 2 ⁢ πF

is the angular frequency (radian/s), F the frequency (Hz), Φ 0

Φ 0 size 12{x rSub { size 8{m} } \( ital "nT" \) } {}

the initial phase (radian), i.e. phase at t = 0. Besides T = 1 F = 2π Ω

1 F = 2π Ω

the period (sec).

We write a similar expression for discrete-time (digital) cosinusoid :

x ( n ) = Acos ⁢ ( ωn + Φ 0 )

(6)

Where A is the amplitude, n the time index, and the quantily ω

ω size 12{ω} {}

will be discussed shortly. For example

x ( n ) = Acos ⁢ ( nπ / 6 + π / 3 )

which is plotted in Figure 4, where the sinusoidal waveshape and the periodicity are very obvious. But it’s not so always (see later).

Figure 4: Signal x(n)= cos ( nπ / 6 + π / 3 )

cos ( nπ / 6 + π / 3 ) size 12{"cos" \( nπ/6 \) +π/3} {}

Comparing (Equation 5) and (Equation 6) we have the following very fundamental relation:

ω = Ω T s

ω = Ω T s size 12{ω= %OMEGA T rSub { size 8{s} } } {}

(7)

The unit of ω

ω size 12{ω} {}

is (radians/s) (s) = radian, but usually interpreted as radians/sample. ω

ω size 12{ω} {}

is called digital angular frequency. We can also defined ω=2πf

ω=2πf size 12{nω=2π} {}

with f the digital frequency (cycles/sample). The digital sinusoid completes a cycle when

nω=2π

nω=2π size 12{nω=2π} {}

radians

ω=2πn

ω=2πn size 12{ω= { {2π} over {n} } } {}

radians/sample

Hence ω

ω size 12{ω} {}

can be considered as the angle extended by two consecutive samples when the samples are uniformly distributed on a circle whose center is the origin.

Because Ω=2πF

Ω=2πF size 12{ %OMEGA =2πF} {}

and Ts=1/fs

Ts=1/fs size 12{T rSub { size 8{s} } =1/f rSub { size 8{s} } } {}

we have from (Reference)

ω=2πFfs

ω=2πFfs size 12{ω=2π { {F} over {f rSub { size 8{s} } } } } {}

(8)

f=Ffs

f=Ffs size 12{f= { {F} over {f rSub { size 8{s} } } } } {}

(9)

(Remember: F is the analog frequency in Hz, fs

fs size 12{f rSub { size 8{s} } } {}

the sampling frequency in samples/cycle, f the digital frequency in samples/cycle). Notice that the digital frequencies ω

ω size 12{ω} {}

and f depend on both the analog frequency F and the sampling frequency fs

fs size 12{f rSub { size 8{s} } } {}

. Notice also that both ω

ω size 12{ω} {}

and f are continuous (only fs

fs size 12{f rSub { size 8{s} } } {}

is discrete). Actually the digital angular frequency ω

ω size 12{ω} {}

is more often used and is called digital frequency for short.

The relation between ω

ω size 12{ω} {}

and F shown in Figure 5. The upper figure shows the relation in linear scale, whereas the lower figure shows the relation on a circle. Remember that the analog frequency F is not periodic, i.e. it can have value between −∞

−∞ size 12{ - infinity } {}

to ∞

∞ size 12{ infinity } {}

while the digital frequency ω

ω size 12{ω} {}

is of circular nature, i.e. it varies along a circle with the periodicity of 2π

2π size 12{2π} {}

, and with the central period from ω=−π

ω=−π size 12{ω= - π} {}

to ω=π

ω=π size 12{ω=π} {}

, corresponding to the Nyquist interval − f s / 2, f s / 2

− f s / 2, f s / 2 size 12{ left [ - f rSub { size 8{s} } /2,f rSub { size 8{s} } /2 right ]} {}

(Fig.1.18). This means that the sinusoids having different frequencies ω

ω size 12{ω} {}

in that interval are separated, while the sinusoids having frequencies outside that interval will be aliased into that interval.

Figure 5: Relation between analog frequency F and digital angular frequency ω

ω size 10{ω} {}

The periodicity of discrete (digital) sinusoids

In Figure 4the signal is cos ( nπ / 6 + π / 3 )

cos ( nπ / 6 + π / 3 ) size 12{"cos" \( nπ/6 \) +π/3} {}

. The envelope of the samples clearly looks both sinusoidal and periodic. The discrete signal is really periodic with the period of 12 samples (by counting the samples and by computing 2π / ( π / 6 )

2π / ( π / 6 ) size 12{2π/ \( π/6 \) } {}

= 12).

There are cases where the discrete signal is periodic but the envelope does not look sinusoidal even if it looks periodic. For example signal x(n)=cos5πn/6

x(n)=cos5πn/6 size 12{x \( n \) ="cos"5πn/6} {}

plotted in Figure 6. The envelope does not bear any shape of a sinusoid, but it looks and it is periodic with a period of 12 samples (by counting, but by computing 2π/(5π/6)

2π/(5π/6) size 12{2π/ \( 5π/6 \) } {}

the result is not 12 ?).

Also there are cases where the samples of a discrete-time signal lie on a sinusoidal and periodic envelope but the signal is not periodic, i.e. the samples do not constitute a periodic sequence. So the

Figure 6: Signal x(n)=cos5πn/6

x(n)=cos5πn/6 size 12{x \( n \) ="cos"5πn/6} {}

periodicity of discrete sinusoids is rather confusing, and we may expect there should be some criterion. For this, let’s begin with a discrete sinusoid cosωn

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DISCRETE TIME SIGNALS

Basic discrete-time signals

(a) Unit sample

(b) Unit step

(c) Unit ramp

(d) Real exponential

1.4.2 Sinusoid, digital frequency, periodicity, complex exponential

The periodicity of discrete (digital) sinusoids