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DISCRETE - TIME SIGNALS

Module by: Nguyen Huu Phuong

As has been said, the samples of a continuous-time signal form the correspronding 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 (uaually 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 as 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)=[−2, −1, 2, −2, −1, 3] x(n)=[...1, −2, 2, 3, 1, −1, 2, −2, 1, 3...] x(n)=[−2, −1, 2, −2, −1, 3] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@77E1@
By this reason, a dicrete-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 corressponding 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 (Fig.1.22a):
δ(n)=1,  n=0    0,  n≠0 δ(n)=1,  n=0    0,  n≠0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOabaeqabaGaeqiTdqMaaiikaiaad6gacaGGPaGaeyypa0JaaGymaiaacYcacaaMf8UaaGzbVlaad6gacqGH9aqpcaaIWaaabaGaaGzbVlaaywW7caaMf8UaaGimaiaacYcacaaMf8UaaGzbVlaad6gacqGHGjsUcaaIWaaaaaa@4EDE@ (1)
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)).
Figure 2: Three basic signals
(b) Unit step
The unit step is defined as (Figure 2 b )
u(n)=1,  n≥0    0,  n<0 (or n<=−1) u(n)=1,  n≥0    0,  n<0 (or n<=−1) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOabaeqabaGaamyDaiaacIcacaWGUbGaaiykaiabg2da9iaaigdacaGGSaGaaGzbVlaaywW7caWGUbGaeyyzImRaaGimaaqaaiaaywW7caaMf8UaaGzbVlaaicdacaGGSaGaaGzbVlaaywW7caWGUbGaeyipaWJaaGimaiaaywW7caGGOaGaam4BaiaadkhacaaMe8UaamOBaiabgYda8iabg2da9iabgkHiTiaaigdacaGGPaaaaaa@5934@ (2)
(c) Unit ramp
This is a divergent signal (amplitude goes to ∞ as n goes to ∞), defined as (Figure 2 c )
r(n)=n,  n≥0    0,  n<0 (or n<=−1) r(n)=n,  n≥0    0,  n<0 (or n<=−1) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOabaeqabaGaamOCaiaacIcacaWGUbGaaiykaiabg2da9iaad6gacaGGSaGaaGzbVlaaywW7caWGUbGaeyyzImRaaGimaaqaaiaaywW7caaMf8UaaGzbVlaaicdacaGGSaGaaGzbVlaaywW7caWGUbGaeyipaWJaaGimaiaaywW7caGGOaGaam4BaiaadkhacaaMe8UaamOBaiabgYda8iabg2da9iabgkHiTiaaigdacaGGPaaaaaa@5969@ (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 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOabaeqabaGaamiEaiaacIcacaWGUbGaaiykaiabg2da9iaadggadaahaaWcbeqaaiaad6gaaaGccaGGSaGaaGzbVlaaywW7caWGUbGaeyyzImRaaGimaaqaaiaaywW7caaMf8UaaGzbVlaaicdacaGGSaGaaGzbVlaaywW7caWGUbGaeyipaWJaaGimaaaaaa@4F88@ (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.

Sinusoid, digital frequency, periodicity, complex exponential

The cosinusoidal signal (Equation 1.1)) is sampled at period TsTs size 12{T rSub { size 8{s} } } {}:
x(t)=Acos⁡(Ωt+ Φ 0 )=Acos⁡(Ωn T s + Φ 0 ) x(t)=Acos⁡(Ωt+ Φ 0 )=Acos⁡(Ωn T s + Φ 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpcaWGbbGaci4yaiaac+gacaGGZbGaaiikaiabfM6axjaadshacqGHRaWkcqqHMoGrdaWgaaWcbaGaaGimaaqabaGccaGGPaGaeyypa0JaamyqaiGacogacaGGVbGaai4CaiaacIcacqqHPoWvcaWGUbGaamivamaaBaaaleaacaWGZbaabeaakiabgUcaRiabfA6agnaaBaaaleaacaaIWaaabeaakiaacMcaaaa@52E8@ (5)
where A is the amplitude, Ω=2πF Ω=2πF MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfM6axjabg2da9iaaikdacqaHapaCcaWGgbaaaa@3BDA@ is the angular frequency (radian/s), F the frequency (Hz), Φo the initial phase (radian), T=1/F=2π/Ω T=1/F=2π/Ω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadsfacqGH9aqpcaaIXaGaai4laiaadAeacqGH9aqpcaaIYaGaeqiWdaNaai4laiabfM6axbaa@3FDA@ the period (sec).
We write a similar expression for discrete-time (digital) cosinusoid :
x(n)=Acos⁡(ωn+ Φ 0 ) x(n)=Acos⁡(ωn+ Φ 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpcaWGbbGaci4yaiaac+gacaGGZbGaaiikaiabeM8a3jaad6gacqGHRaWkcqqHMoGrdaWgaaWcbaGaaGimaaqabaGccaGGPaaaaa@454F@ (6)
Where A is the amplitude, n the time index , and the quantily ωω size 12{ω} {} will be discussed shortly. For example
x(n)=Acos⁡(nπ/6+π/3) x(n)=Acos⁡(nπ/6+π/3) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpcaWGbbGaci4yaiaac+gacaGGZbGaaiikaiaad6gacqaHapaCcaGGVaGaaGOnaiabgUcaRiabec8aWjaac+cacaaIZaGaaiykaaaa@4775@
which is plotted in Fig.1.24, where the sinusoidal waveshape and the periodicity are very obvious. But it’s not so always (see later).
Figure 4: Signal x(n)=cos⁡( π 6 n+ π 3 ) x(n)=cos⁡( π 6 n+ π 3 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpciGGJbGaai4BaiaacohacaGGOaWaaSaaaeaacqaHapaCaeaacaaI2aaaaiaad6gacqGHRaWkdaWcaaqaaiabec8aWbqaaiaaiodaaaGaaiykaaaa@4569@
Comparing (Equation 5) and (Equation 6) we have the following very fundamental relation:
ω=Ω T s ω=Ω T s MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3jabg2da9iabfM6axjaadsfadaWgaaWcbaGaam4Caaqabaaaaa@3C60@ (7)
The unit of ωω size 12{ω} {}is (radian/s) (s) = radian, but usually interpreted as radians/sample . ωω size 12{ω} {} is called digital angular frequency . We can also defined ωω size 12{ω} {}= 2f with f the digital frequency (cycles/sample). The digital sinusoid completes a cycle when
nω=2π nω=2π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaad6gacqaHjpWDcqGH9aqpcaaIYaGaeqiWdahaaa@3C41@
or
ω= 2π n ω= 2π n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3jabg2da9maalaaabaGaaGOmaiabec8aWbqaaiaad6gaaaaaaa@3C51@
Hence ωω 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=2πFΩ=2πF size 12{ %OMEGA =2πF} {}and Ts=1/fsTs=1/fs size 12{T rSub { size 8{s} } =1/f rSub { size 8{s} } } {} we have from Equation (Equation 7)
ω=2πFfsω=2πFfs size 12{ω="2π" { {F} over {f rSub { size 8{s} } } } } {}(8)
or
f=Ffsf=Ffs size 12{f= { {F} over {f rSub { size 8{s} } } } } {} (9)
(Remember: F is the analog frequency in Hz, fsfs 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 ωω size 12{ω} {}and f depend on both the analog frequency F and the sampling frequency fsfs size 12{f rSub { size 8{s} } } {}. Notice also that both ωω size 12{ω} {} and f are continuous (only fsfs size 12{f rSub { size 8{s} } } {} is discrete). Actually the digital angular frequency ωω size 12{ω} {} is more often used and is called digital frequency for short. However some authors prefer using f.
The relation between ωω 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 ωω size 12{ω} {}is of circular nature, i.e. it varies along a circle with the periodicity of 2π2π size 12{2π} {}, and with the central period from ω=−πω=−π size 12{ω= - π} {} to ω=πω=π size 12{ω=π} {}, corresponding to the Nyquist interval [-fs/2, fs/2] (Fig.1.18). This means that the sinusoids having different frquencies ωω 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 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3baa@37CF@
The periodicity of discrete (digital) sinusoids
In Figure 4 the signal is cos⁡(πn/6+π/3) cos⁡(πn/6+π/3) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiGacogacaGGVbGaai4CaiaacIcacqaHapaCcaWGUbGaai4laiaaiAdacqGHRaWkcqaHapaCcaGGVaGaaG4maiaacMcaaaa@4260@ . 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)=12 2π/(π/6)=12 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaaikdacqaHapaCcaGGVaGaaiikaiabec8aWjaac+cacaaI2aGaaiykaiabg2da9iaaigdacaaIYaaaaa@4034@ .
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/6x(n)=cos5πn/6 size 12{x \( n \) ="cos"5πn/6} {} plotted in Fig.1.26. 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 sinưsoidal 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)=cos⁡(5πn/6) x(n)=cos⁡(5πn/6) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqabeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpciGGJbGaai4BaiaacohacaGGOaGaaGynaiabec8aWjaad6gacaGGVaGaaGOnaiaacMcaaaa@435F@
So, the 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 cosωncosωn size 12{"cos"ωn} {} periodic with a period of N samples:
cos ωn = cos ω ( n + N ) = cos ( ωn + ωN ) cos ωn = cos ω ( n + N ) = cos ( ωn + ωN ) size 12{"cos"ωn="cos"ω \( n+N \) ="cos" \( ωn+ωN \) } {}
The condition is that ωNωN size 12{ωN} {}must be some integer m of 2π2π size 12{2π} {}:
ωN=2πm,  m integers ωN=2πm,  m integers MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3jaad6eacqGH9aqpcaaIYaGaeqiWdaNaamyBaiaacYcacaaMf8UaaGzbVlaad2gacaaMf8ocbiGaa8xAaiaa=5gacaWF0bGaamyzaiaadEgacaWGLbGaamOCaiaadohaaaa@4AE9@
or
ω2π=f=mNω2π=f=mN size 12{ { {ω} over {2π} } =f= { {m} over {N} } } {}(10)
This means that ω/2πω/2π size 12{ω/2π} {} (or f) must be a rational function (ratio of two integers). The actual period N will equal to the denomitator of m/N after simplification (cancelling out the common factor). If ω/2πω/2π size 12{ω/2π} {} is not a rational number the signal is not periodic (i.e. nonperiodic or aperiodic). As said earlier, even though a discrete sinusoid may or may not periodic but its envelope of samples is always periodic (but may not be sinusoidal). Following is some examples.
  • Signal cos(πn/6)cos(πn/6) size 12{"cos" \( πn/6 \) } {} is periodic because ω/2π=(π/6)/2π=1/12ω/2π=(π/6)/2π=1/12 size 12{ω/2π= \( π/6 \) /2π=1/"12"} {} (a rational function) and the period is 12.
  • Signal cos5πn/6cos5πn/6 size 12{"cos"5πn/6} {} is periodic because ω/2π=(5π/6)/2π=5/12ω/2π=(5π/6)/2π=5/12 size 12{ω/2π= \( 5π/6 \) /2π=5/"12"} {}(a rational function) and the period is 12 (the period is not 2π/(5π/6)2π/(5π/6) size 12{2π/ \( 5π/6 \) } {}as for analog sinusoid).
  • Signal cosπn/8cosπn/8 size 12{"cos"πn/8} {} is periodic with period = 16, whereas cos0.4n is not ( ω/2π=0.4/2πω/2π=0.4/2π size 12{ω/2π=0 "." 4/2π} {} is not a rational function). We can plot out these two signals the first 30 samples to check.
A point about periodicity of discrete sinusoid should be added before we leave the topic: A small change in digital frequency can lead to a large change in period. For example, with ω/2π=51/100ω/2π=51/100 size 12{ω/2π="51"/"100"} {} the period is 100, but with ω/2π = 50/100 = 1/2, the period is just 2.
Complex exponential
We are considering the signals of the type x(n)= a n x(n)= a n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpcaWGHbWaaWbaaSqabeaacaWGUbaaaaaa@3C59@ with constant a complex, called complex exponential, or complex sinusoid by some authors. Let
a=r e jω a=r e jω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadggacqGH9aqpcaWGYbGaamyzamaaCaaaleqabaGaamOAaiabeM8a3baaaaa@3CBA@ (11)
then the signal is
x(n)= (r e jω ) n = r n e jωn = 2 n (cos⁡ωn+jsin⁡ωn) x(n)= (r e jω ) n = r n e jωn = 2 n (cos⁡ωn+jsin⁡ωn) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhacaGGOaGaamOBaiaacMcacqGH9aqpcaGGOaGaamOCaiaadwgadaahaaWcbeqaaiaadQgacqaHjpWDaaGccaGGPaWaaWbaaSqabeaacaWGUbaaaOGaeyypa0JaamOCamaaCaaaleqabaGaamOBaaaakiaadwgadaahaaWcbeqaaiaadQgacqaHjpWDcaWGUbaaaOGaeyypa0JaaGOmamaaCaaaleqabaGaamOBaaaakiaacIcaciGGJbGaai4BaiaacohacqaHjpWDcaWGUbGaey4kaSIaamOAaiGacohacaGGPbGaaiOBaiabeM8a3jaad6gacaGGPaaaaa@5AE2@ (12)
whose real and imginary components are, respectively,
x R (n)= r n cos⁡nω x I (n)= r n sin⁡nω x R (n)= r n cos⁡nω x I (n)= r n sin⁡nω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOabaeqabaGaamiEamaaBaaaleaacaWGsbaabeaakiaacIcacaWGUbGaaiykaiabg2da9iaadkhadaahaaWcbeqaaiaad6gaaaGcciGGJbGaai4BaiaacohacaWGUbGaeqyYdChabaGaamiEamaaBaaaleaacaWGjbaabeaakiaacIcacaWGUbGaaiykaiabg2da9iaadkhadaahaaWcbeqaaiaad6gaaaGcciGGZbGaaiyAaiaac6gacaWGUbGaeqyYdChaaaa@5027@
From these we get the magnitude and phase:
|x(n)|= x R 2 (n)+ x I 2 (n) = r n |x(n)|= x R 2 (n)+ x I 2 (n) = r n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacYhacaWG4bGaaiikaiaad6gacaGGPaGaaiiFaiabg2da9maakaaabaGaamiEamaaDaaaleaacaWGsbaabaGaaGOmaaaakiaacIcacaWGUbGaaiykaiabgUcaRiaadIhadaqhaaWcbaGaamysaaqaaiaaikdaaaGccaGGOaGaamOBaiaacMcaaSqabaGccqGH9aqpcaWGYbWaaWbaaSqabeaacaWGUbaaaaaa@4A94@
Φ(n)= tan⁡ −1 x I (n) x R (n) = tan⁡ −1 (tgnω)=nω Φ(n)= tan⁡ −1 x I (n) x R (n) = tan⁡ −1 (tgnω)=nω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfA6agjaacIcacaWGUbGaaiykaiabg2da9iGacshacaGGHbGaaiOBamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaalaaabaGaamiEamaaBaaaleaacaWGjbaabeaakiaacIcacaWGUbGaaiykaaqaaiaadIhadaWgaaWcbaGaamOuaaqabaGccaGGOaGaamOBaiaacMcaaaGaeyypa0JaciiDaiaacggacaGGUbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaiikaiaadshacaWGNbGaamOBaiabeM8a3jaacMcacqGH9aqpcaWGUbGaeqyYdChaaa@57AD@
Actually, from Equation 12 we can see straightaway these results.
Figure 7: Example 1.4.1
Example 1 
Plot x R (n) x R (n) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaamOuaaqabaGccaGGOaGaamOBaiaacMcaaaa@3A5A@ , x I (n) x I (n) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaadIhadaWgaaWcbaGaamysaaqabaGccaGGOaGaamOBaiaacMcaaaa@3A51@ , │x(n)│, and Φ(n) when r = 0.9 and ω=π/10 ω=π/10 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabeM8a3jabg2da9iabec8aWjaac+cacaaIXaGaaGimaaaa@3CBC@ .
Solution
The necessary expressions are
x ( n ) = 0,9 n e jπn / 10 x ( n ) = 0,9 n e jπn / 10 size 12{x \( n \) =0,9 rSup { size 8{n} } e rSup { size 8{jπn/"10"} } } {}
x R ( n ) = 0,9 n cos πn / 10 x R ( n ) = 0,9 n cos πn / 10 size 12{x rSub { size 8{R} } \( n \) =0,9 rSup { size 8{n} } "cos"πn/"10"} {}
x I ( n ) = 0,9 n sin πn / 10 x I ( n ) = 0,9 n sin πn / 10 size 12{x rSub { size 8{I} } \( n \) =0,9 rSup { size 8{n} } "sin"πn/"10"} {}
∣ x ( n ) ∣ = 0,9 n ∣ x ( n ) ∣ = 0,9 n size 12{ lline x \( n \) rline =0,9 rSup { size 8{n} } } {}
Φ ( n ) = πn / 10 Φ ( n ) = πn / 10 size 12{Φ \( n \) =πn/"10"} {}
Fig.1.41 shows the results. Notice, especially, the phase response Φ(n) Φ(n) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabfA6agjaacIcacaWGUbGaaiykaaaa@39CA@ . The phase response is by convention, limited in the range [−π,π] [−π,π] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiaacUfacqGHsislcqaHapaCcaGGSaGaeqiWdaNaaiyxaaaa@3CDB@ . At n = 10, the phase is π π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabec8aWbaa@37C1@ −π −π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipDI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeaabiGaciaacaqaaeaadaqaaqaaaOqaaiabgkHiTiabec8aWbaa@38AE@

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