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EEEN 214

Signals and Systems

Lecture Notes 06

Properties of the Fourier Series

Let x(t) and y(t) be periodic signals with periods T and Fourier Series representations 𝑥(𝑡) 𝑎𝑘 𝑦(𝑡) 𝑏𝑘 Then the following properties hold: Linearity Time shift Frequency shift Conjugation Time reversal Time scaling

−𝑗𝑘𝜔 0 𝑡 0 𝑒𝑗𝑀𝜔^0 𝑡𝑥 𝑡 𝑎𝑘−𝑀 𝑥 ∗ 𝑡 𝑎 ∗ −𝑘 𝑥 −𝑡 𝑎−𝑘 𝑥 𝛼𝑡 𝑎𝑘 Periodic convolution Multiplication 𝑥^ 𝑡^ 𝑦(𝑡)^ ෍ 𝑛=−∞ ∞ 𝑎𝑛𝑏𝑘−𝑛 Differentiation

Integration න −∞ 𝑡 𝑥 𝜏 𝑑𝜏

2

𝑇

Review Questions

• Is the Fourier-series representation valid for any type of signal?
• What can you say about the Fourier series coefficients of a real signal? Periodic signals only
• What is the Gibbs phenomenon? Oscillation in Fourier approximation due to signal discontinuity They satisfy the complex conjugation property
• What can you say about the Fourier series coefficients of a real and even signal? They are real and even
• What can you say about the Fourier series coefficients of a real and odd signal? They are purely imaginary and odd

• We have seen that complex exponentials are eigenfunctions of LTI systems:
• By linearity:

𝑠𝑡 𝑦(𝑡) = 𝐻(𝑠) 𝑒 𝑠𝑡 𝑥(𝑡) = 𝑎 1 𝑒 𝑠 1 𝑡

• 𝑎 2 𝑒 𝑠 2 𝑡
• 𝑎 3 𝑒 𝑠 3 𝑡 𝑦(𝑡) = 𝑎 1 𝐻(𝑠 1 ) 𝑒 𝑠 1 𝑡
• 𝑎 2 𝐻(𝑠 2 ) 𝑒 𝑠 2 𝑡
• 𝑎 3 𝐻(𝑠 3 ) 𝑒 𝑠 3 𝑡 𝑥(𝑡) = ෍ 𝑘

𝑠𝑘𝑡 𝑦(𝑡) = ෍ 𝑘

𝑠𝑘𝑡

• In general:

−∞ ∞ ℎ(𝜏)𝑒 −𝑠𝜏 𝑑𝜏

• Letting 𝑠𝑘 = 𝑗𝑘𝜔 0 we have: 𝑦(𝑡) = ෍ 𝑘

𝑗𝑘𝜔 0 𝑡 𝑥(𝑡) = ෍ h(t) 𝑘

𝑗𝑘𝜔 0 𝑡 h(t) h(t) h(t)

Fourier Series and LTI Systems

Fourier Series and LTI Systems (cnt’d)

• Letting 𝑠 = 𝑗𝜔 in 𝐻(𝑠) = න −∞ ∞ ℎ(𝜏)𝑒 −𝑠𝜏 𝑑𝜏 𝐻(𝑗𝜔) = න −∞ ∞ ℎ(𝑡)𝑒 −𝑗𝜔𝑡 𝑑𝑡
• We will see later that ℎ(𝑡) and 𝐻(𝑗𝜔) form a Fourier-transform pair.

yields:

system frequency response

system transfer function

• Note that the integral must converge for the system frequency response to exist

[not all choices of ℎ(𝑡) result in convergence for the above integral].

Stability

• For the expression σ𝑘 𝑎𝑘𝐻(𝑗𝑘𝜔 0 )𝑒 𝑗𝑘𝜔 0 𝑡

to make sense, the frequency response

𝐻(𝑗𝜔) must be well-defined and finite:

• This will be the case if the LTI system considered is stable.
• For example, the LTI system with impulse response ℎ 𝑡 = 𝑒 −𝑡

𝑢(𝑡) is stable and

has a well defined frequency response 𝐻 𝑗𝜔 = 1 Τ ( 1 + 𝑗𝜔).

• On the other hand, the system with impulse response ℎ 𝑡 = 𝑒 𝑡

𝑢(𝑡) is unstable

because the computation of 𝐻 𝑗𝜔 diverges.

−∞ ∞ ℎ(𝑡)𝑒 −𝑗𝜔𝑡 𝑑𝑡

𝑘

𝑗𝑘𝜔 0 𝑡 𝑥(𝑡) = ෍ h(t) 𝑘

𝑗𝑘𝜔 0 𝑡

cos(𝑘𝜔 0 𝑡) h(t) 𝛼 cos(𝑘𝜔 0 𝑡 + 𝜃)

Exercise

• Assume that the input of an LTI system with impulse response ℎ 𝑡 is cos(𝑘𝜔 0 𝑡).

Show that the output of this system is 𝛼 cos(𝑘𝜔 0 𝑡 + 𝜃), where 𝛼 = 𝐻(𝑗𝜔 0 ) and

𝜃 = ∠𝐻(𝑗𝜔 0 ), if 𝐻 𝑗𝜔 = 𝐻

∗

−𝑗𝜔 [as we will see later, this implies real-valued

ℎ 𝑡 ].