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CONTINUOUS-TIME SIGNALS

Module by: Nguyen Huu Phuong

Signals is the variation of an amplitude with time. The amplitude can be voltage, current, power,... But in circuits and systems the most often used representative is voltage. Thus the two elements constituting signals are amplitude and time.
Continuous – time (also taken to mean analog) signals have their amplitudes varying continuously with time. They are generated by electronic circuits, or by natural sources, such as temperature, voice, video..., and converted to electric signals by sensors or transducers. Signals are often depicted by their waveforms which are the graphical illustrations for easy visualization.

Mathematical representation of signals

Instead of describing signals by words or by plotting their waveforms, the more objective and concise way is to express them mathematically, whenever possible. Mathematical representation of signals in time domain and transform domain is needed in analysis and design of circuits and systems. For example a simple problem in Figure 1 cannot be solved just by, language description of the signal and the circuit.
Figure 1: What is the output signal ?
Sinusoidal signal
Sinusoid or sinewave is the most popular analog signal (Figure 2). It is smooth, easy to generate, and has many properties and applications. The mathematical expression is
x(t)=Acos⁡(Ωt+ Φ 0 ) x(t)=Acos⁡(Ωt+ Φ 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpcaWGbbGaci4yaiaac+gacaGGZbGaaiikaiabfM6axjaadshacqGHRaWkcqqHMoGrdaWgaaWcbaGaaGimaaqabaGccaGGPaaaaa@4500@ (1)
Figure 2: Sinusoidal signal
where A is the peak value, Ω angular frequency (rad/s), t time (sec), Φ 0 Φ 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabfA6agnaaBaaaleaacaaIWaaabeaaaaa@3846@ initial phase (radian) i.e. phase at t = 0, Ω=2πF Ω=2πF MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabfM6axjabg2da9iaaikdacqaHapaCcaWGgbaaaa@3BBE@ with F is frequency (Hz)(Hz) size 12{ \( "Hz" \) } {}, T = 1 / F=2π/Ω F=2π/Ω MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeacqGH9aqpcaaIYaGaeqiWdaNaai4laiabfM6axbaa@3C71@ the period (sec).
Above expression contains all parameters we need: Amplitude (peak, rms, average), and periodicity (period, frequency). Other waveforms, except the constant value, do not have this compactness. For example, for the symmetric square wave (Figure 3) the mathematical expression consists of one part for amplitude, and the other for periodicity:
x(t)=−A ,  − T 2 ≤t≤0    +A ,  0≤t≤ T 2 x(t)=x(t±nT) , n=1, 2, 3... x(t)=−A ,  − T 2 ≤t≤0    +A ,  0≤t≤ T 2 x(t)=x(t±nT) , n=1, 2, 3... MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@7556@ (2)
The sinusoid and the square wave are deterministic . For random signals, we cannot, in general, represent them mathematically. Electric noises and interferences are examples of random signals.
Figure 3: Symmetric square wave

Some special signals

There are two singular signals often used in circuit analysis and signal processing.
(a) Unit impulse:
The unit impulse (delta Dirac function) is evolved from a symmetric rectangular pulse of width ττ size 12{τ} {} and amplitude 1/τ1/τ size 12{1/τ} {} when ττ size 12{τ} {}→ 0 (Figure 4). Its mathematical expression is
δ(t)=∞ ,  t=0    0 ,  t≠0 ∫ −∞ ∞ δ(t)dt=1 δ(t)=∞ ,  t=0    0 ,  t≠0 ∫ −∞ ∞ δ(t)dt=1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeqiTdqMaaiikaiaadshacaGGPaGaeyypa0JaeyOhIuQaaGzbVlaacYcacaaMf8UaaGzbVlaadshacqGH9aqpcaaIWaaabaaabaGaaGzbVlaaywW7caaMf8UaaGimaiaaywW7caGGSaGaaGzbVlaaywW7caWG0bGaeyiyIKRaaGimaaqaaaqaamaapedabaGaeqiTdqMaaiikaiaadshacaGGPaGaamizaiaadshacqGH9aqpcaaIXaaaleaacqGHsislcqGHEisPaeaacqGHEisPa0Gaey4kIipaaaaa@603B@ (3)
According to this definition,
δ(−t)=δ(t) δ(−t)=δ(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaacIcacqGHsislcaWG0bGaaiykaiabg2da9iabes7aKjaacIcacaWG0bGaaiykaaaa@3FC7@ (4)
Figure 4: Unit impulse δ(t) δ(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaacIcacaWG0bGaaiykaaaa@39DD@
When the impulse has intensity of A instead of 1 we write Aδ(t) Aδ(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeacqaH0oazcaGGOaGaamiDaiaacMcaaaa@3AA3@ (Figure 4c). When the unit impulse is delayed by t 0 t 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadshadaWgaaWcbaGaaGimaaqabaaaaa@37C5@ we write δ(t− t 0 ) δ(t− t 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaacIcacaWG0bGaeyOeI0IaamiDamaaBaaaleaacaaIWaaabeaakiaacMcaaaa@3CB3@ (Figure 4d), then
δ(t− t 0 )=∞ ,  t= t 0       0 ,  t≠ t 0 δ(t− t 0 )=∞ ,  t= t 0       0 ,  t≠ t 0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeqiTdqMaaiikaiaadshacqGHsislcaWG0bWaaSbaaSqaaiaaicdaaeqaaOGaaiykaiabg2da9iabg6HiLkaaywW7caGGSaGaaGzbVlaaywW7caWG0bGaeyypa0JaamiDamaaBaaaleaacaaIWaaabeaaaOqaaaqaaiaaywW7caaMf8UaaGzbVlaaywW7caaMe8UaaGPaVlaaicdacaaMf8UaaiilaiaaywW7caaMf8UaamiDaiabgcMi5kaadshadaWgaaWcbaGaaGimaaqabaaaaaa@5C77@
∫ −∞ ∞ δ(t− t 0 )dt = ∫ t 1 t 2 δ(t− t 0 )dt=1 ,   t 1 < t 0 < t 2 ∫ −∞ ∞ δ(t− t 0 )dt = ∫ t 1 t 2 δ(t− t 0 )dt=1 ,   t 1 < t 0 < t 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@630F@ (5)
A signal x(t) when multiplied with delayed impulse δ(t−t0)δ(t−t0) size 12{δ \( t - t rSub { size 8{0} } \) } {}is the value x(to) at to:
x(t)δ(t− t 0 )=x( t 0 ) x(t)δ(t− t 0 )=x( t 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqaH0oazcaGGOaGaamiDaiabgkHiTiaadshadaWgaaWcbaGaaGimaaqabaGccaGGPaGaeyypa0JaamiEaiaacIcacaWG0bWaaSbaaSqaaiaaicdaaeqaaOGaaiykaaaa@4547@ (6)
(b) Unit step:
Figure 5 is the unit step. The signal rises suddenly from 0 to 1 at time t = 0 then remains unchanged, similarly to the closure of an electric switch. Its mathematical definition is
u(t)=0 ,  t<0      1 ,  t≥0 u(t)=0 ,  t<0      1 ,  t≥0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamyDaiaacIcacaWG0bGaaiykaiabg2da9iaaicdacaaMf8UaaiilaiaaywW7caaMf8UaamiDaiabgYda8iaaicdaaeaaaeaacaaMf8UaaGzbVlaaysW7caaMe8UaaGjbVlaaigdacaaMf8UaaiilaiaaywW7caaMf8UaamiDaiabgwMiZkaaicdaaaaa@545C@ (7)
Figure 5: Unit step
The unit impulse δ(t) δ(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabes7aKjaacIcacaWG0bGaaiykaaaa@39DD@ and the unit step u(t) are related as follows.
u(t)= ∫ δ( t , )d t , =0 ,   t<0         1 ,  t≥0 u(t)= ∫ δ( t , )d t , =0 ,   t<0         1 ,  t≥0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamyDaiaacIcacaWG0bGaaiykaiabg2da9maapedabaGaeqiTdqMaaiikaiaadshadaahaaWcbeqaaiaacYcaaaGccaGGPaGaamizaiaadshadaahaaWcbeqaaiaacYcaaaaabaaabaaaniabgUIiYdGccqGH9aqpcaaIWaGaaGzbVlaacYcacaaMf8UaaGjcVlaaywW7caWG0bGaeyipaWJaaGimaaqaaaqaaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caaMf8UaaGymaiaaywW7caGGSaGaaGzbVlaaywW7caWG0bGaeyyzImRaaGimaaaaaa@6565@ (8)
δ(t)=du(t)dtδ(t)=du(t)dt size 12{δ \( t \) = { { ital "du" \( t \) } over { ital "dt"} } } {} (9)

Complex signals

Natural physical quantities, signals included, are real-valued. However sometimes the imaginary operator j = −1−1 size 12{ sqrt { - 1} } {} is appended by reason of mathematical convenience, such as to take into account the phase difference between voltages and currents in AC circuits. Following is an example of a complex signal :
x(t)=5cos⁡Ωt−j5cos⁡Ωt x(t)=5cos⁡Ωt−j5cos⁡Ωt MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpcaaI1aGaci4yaiaac+gacaGGZbGaeuyQdCLaamiDaiabgkHiTiaadQgacaaI1aGaci4yaiaac+gacaGGZbGaeuyQdCLaamiDaaaa@4849@ (10)
A complex signal comprises of a real part and an imaginary one:
x(t)= x R (t)+j x I (t) x(t)= x R (t)+j x I (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpcaWG4bWaaSbaaSqaaiaadkfaaeqaaOGaaiikaiaadshacaGGPaGaey4kaSIaamOAaiaadIhadaWgaaWcbaGaamysaaqabaGccaGGOaGaamiDaiaacMcaaaa@44BB@ (11)
A complex signal can be expressed in terms of its magnitude and phase in the polar coordinate (Figure 6)
x(t)= x R (t)+j x I (t)=|x(t)| e jΦ(t) x(t)= x R (t)+j x I (t)=|x(t)| e jΦ(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcacqGH9aqpcaWG4bWaaSbaaSqaaiaadkfaaeqaaOGaaiikaiaadshacaGGPaGaey4kaSIaamOAaiaadIhadaWgaaWcbaGaamysaaqabaGccaGGOaGaamiDaiaacMcacqGH9aqpcaGG8bGaamiEaiaacIcacaWG0bGaaiykaiaacYhacaWGLbWaaWbaaSqabeaacaWGQbGaeuOPdyKaaiikaiaadshacaGGPaaaaaaa@50E2@ (12)
Figure 6: Complex signal and polar coordinate
The magnitude or modulus, demoted by ∣x(t)∣∣x(t)∣ size 12{ lline x \( t \) rline } {} , and phase or phase angle, demoted by Φ(t)Φ(t) size 12{Φ \( t \) } {}or argx(t) or ∠∠ size 12{∠} {} x(t). They are
∣ x ( t ) ∣ = x R 2 ( t ) + x I 2 ( t ) ∣ x ( t ) ∣ = x R 2 ( t ) + x I 2 ( t ) size 12{ lline x \( t \) rline = nroot {} {x rSub { size 8{R} } rSup { size 8{2} } \( t \) +x rSub { size 8{I} } rSup { size 8{2} } \( t \) } } {}
Φ ( t ) = tan − 1 x I ( t ) x R ( t ) Φ ( t ) = tan − 1 x I ( t ) x R ( t ) size 12{Φ \( t \) ="tan" rSup { size 8{ - 1} } { {x rSub { size 8{I} } \( t \) } over {x rSub { size 8{R} } \( t \) } } } {}
A point to note is that magnitude is an absolute value while amplitude is a signed value, but we do not always need to differentiate the two terms.
Example 1 
A complex signal is as in Equation 10. Find its real part, imaginary part, magnitude and phase.
Solution
  • Real part: x RR size 12{ {} rSub { size 8{R} } } {}(t) = 5cos ΩΩ size 12{ %OMEGA } {}t
  • Imaginary part: x II size 12{ {} rSub { size 8{I} } } {}(t) = –5cos ΩΩ size 12{ %OMEGA } {}t
  • Magnitude: ∣x(t)∣=(5cosΩt)2+(−5cosΩt)21/2=52cosΩt∣x(t)∣=(5cosΩt)2+(−5cosΩt)21/2=52cosΩt size 12{ lline x \( t \) rline = left [ \( "5cos" %OMEGA t \) rSup { size 8{2} } + \( - "5cos" %OMEGA t \) rSup { size 8{2} } right ] rSup { size 8{1/2} } =5 sqrt {2} "cos" %OMEGA t} {}
  • Phase: Φ(t)=tan−1−5cosΩt5cosΩt=tan−1(−1)Φ(t)=tan−1−5cosΩt5cosΩt=tan−1(−1) size 12{Φ \( t \) ="tan" rSup { size 8{ - 1} } { { - 5"cos" %OMEGA t} over {5"cos" %OMEGA t} } ="tan" rSup { size 8{ - 1} } \( - 1 \) } {}= –450 independent of t
According to this representation we can consider a complex signal as a vector and write x(t).
Figure 7: A complex signal x(t) x(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcaaaa@3935@ and its complex conjugate x * (t) x * (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaahaaWcbeqaaiaacQcaaaGccaGGOaGaamiDaiaacMcaaaa@3A1A@
Two complex quantities having the same real part but opposile imaginary part are complex conjugates of each other (Figure 7). Thus for a given complex signal x(t) in Equation 12, its complex conjugate is
x * (t)= x R (t)−j x I (t)=|x(t)| e −jΦ(t) x * (t)= x R (t)−j x I (t)=|x(t)| e −jΦ(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaahaaWcbeqaaiaacQcaaaGccaGGOaGaamiDaiaacMcacqGH9aqpcaWG4bWaaSbaaSqaaiaadkfaaeqaaOGaaiikaiaadshacaGGPaGaeyOeI0IaamOAaiaadIhadaWgaaWcbaGaamysaaqabaGccaGGOaGaamiDaiaacMcacqGH9aqpcaGG8bGaamiEaiaacIcacaWG0bGaaiykaiaacYhacaWGLbWaaWbaaSqabeaacqGHsislcaWGQbGaeuOPdyKaaiikaiaadshacaGGPaaaaaaa@52BF@ (13)

Complex exponential signals

Equation 1 is a real sinusoidal signal. Complex exponentials, also called complex sinusoids, are more often used. The general expression is
x(t)=Aej(Ωt+Φ0)x(t)=Aej(Ωt+Φ0) size 12{x \( t \) = ital "Ae" rSup { size 8{j \( %OMEGA t+Φ rSub { size 6{0} } \) } } } {} (14)
Phasor is the vector representation of the signal (Figure 8). It is periodic with an angular period of 2 π π MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiabec8aWbaa@37A3@ radians.
Figure 8: Phasor representing complex exponential
From a complex exponential, its real sinusoidal part is deduced by two ways. The first is to take the real part:
x R (t)=Re⁡[ Acos⁡(Ωt+ Φ 0 )+jAsin⁡(Ωt+ Φ 0 ) ]     =Acos⁡(Ωt+ Φ 0 ) x R (t)=Re⁡[ Acos⁡(Ωt+ Φ 0 )+jAsin⁡(Ωt+ Φ 0 ) ]     =Acos⁡(Ωt+ Φ 0 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamiEamaaBaaaleaacaWGsbaabeaakiaacIcacaWG0bGaaiykaiabg2da9iGackfacaGGLbWaamWaaeaacaWGbbGaci4yaiaac+gacaGGZbGaaiikaiabfM6axjaadshacqGHRaWkcqqHMoGrdaWgaaWcbaGaaGimaaqabaGccaGGPaGaey4kaSIaamOAaiaadgeaciGGZbGaaiyAaiaac6gacaGGOaGaeuyQdCLaamiDaiabgUcaRiabfA6agnaaBaaaleaacaaIWaaabeaakiaacMcaaiaawUfacaGLDbaaaeaaaeaacaaMf8UaaGzbVlaaysW7caaMc8Uaeyypa0JaamyqaiGacogacaGGVbGaai4CaiaacIcacqqHPoWvcaWG0bGaey4kaSIaeuOPdy0aaSbaaSqaaiaaicdaaeqaaOGaaiykaaaaaa@6862@ (15)
Figure 9: adding a phasor x(t) x(t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhacaGGOaGaamiDaiaacMcaaaa@3935@ with its complex conjugate to form the real part 2 x R (t) 2 x R (t) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqadeqabiGaciaacaqabeaadaqaaqaaaOqaaiaaikdacaWG4bWaaSbaaSqaaiaadkfaaeqaaOGaaiikaiaadshacaGGPaaaaa@3AFE@
This is just the projection the phasor onto the real axis. The second way is to use two phasors, x(t) and its complex conjugate x*(t), and then take half of the sum:
x R (t)= 1 2 [ x(t)+ x * (t) ]     = 1 2 [ A e j( Ω 0 t+ Φ 0 ) +A e −j( Ω 0 t+ Φ 0 ) ] x R (t)= 1 2 [ x(t)+ x * (t) ]     = 1 2 [ A e j( Ω 0 t+ Φ 0 ) +A e −j( Ω 0 t+ Φ 0 ) ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@699C@ (16)
Notice that when the two phasors rotate in opposite directions at angular frequencies Ω and −Ω−Ω size 12{ - %OMEGA } {}, the addition always gives twice the real sinusoid.

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