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Lecture 5 ELE 301: Signals and Systems Prof. Paul Cuff Princeton University Fall 2011-12 Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 1 / 24
Lecture 5ELE 301: Signals and Systems
Prof. Paul Cuff
Princeton University
Fall 2011-12
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 1 / 24
History of the Fourier Series
Euler (1748): Vibrations of a string
Fourier: Heat dynamics
Dirichlet (1829): Convergence of the Fourier Series
Lagrange: Rejected publication
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 2 / 24
What is the Fourier Series
The Fourier Series allows us to represent periodic signals as sums ofsinusoids.
x(t) =∞∑
k=−∞ake
jk2πf0t
where f0 = 1/T0 and T0 is the fundamental period.
There are other transforms for representing signalsI Wavelet transformI Taylor expansionI Any orthonormal basis
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 3 / 24
Response of LTI Systems to Exponential FunctionsFor an LTI system with impulse response h(t), output is the convolution ofinput and impulse response:
y(t) =
∫ ∞−∞
h(τ)x(t − τ) dτ
x(t) y(t)∗h(t)
If the input is a complex exponential x(t) = e jωt
y(t)∗h(t)
e jωt
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 4 / 24
Eigenfunctions
Continuous time:est −→h H(s)est
Discrete time:zn −→h H(z)zn
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 5 / 24
Aliasing
Wolfram Demo:
e(σ+j(2πf ))n = e(σ+j(2π(f +k)))n for all integers n and k .
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 6 / 24
Sums of Exponentials
a1 +es1t +a2 +es2t +a3 +es3t −→h a1H(s1)es1t +a2H(s2)es2t +a3H(s3)es3t
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 7 / 24
Period Signals
Claim:
x(t) =∞∑
k=−∞ake
jk2πf0t
where f0 = 1/T0 and T0 is the fundamental period.
Consider an easy one:
x(t) = cos(2πf0t)
=1
2e2πf0t +
1
2e−2πf0t .
Therefore, T = 1/f0 and a1 = a−1 = 1/2.
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 8 / 24
Fourier Series approximation to a square wave
!2 !1.5 !1 !0.5 0 0.5 1 1.5 2!0.2
0
0.2
0.4
0.6
0.8
1
1.2
!2 !1.5 !1 !0.5 0 0.5 1 1.5 2!0.2
0
0.2
0.4
0.6
0.8
1
1.2
2 Terms
4 Terms
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 9 / 24
Fourier Series approximation to a square wave
!2 !1.5 !1 !0.5 0 0.5 1 1.5 2!0.2
0
0.2
0.4
0.6
0.8
1
1.2
!2 !1.5 !1 !0.5 0 0.5 1 1.5 2!0.2
0
0.2
0.4
0.6
0.8
1
1.2
8 Terms
16 Terms
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 10 / 24
Real Signals
If x is real
x(t) = a0 + 2∞∑k=1
Ak cos(k2πf0t + θk),
where Akejθk = ak .
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 11 / 24
Fourier Series Coefficients
ak =1
T
∫Tx(t)e−jk2πf0tdt.
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 12 / 24
Conditions for Convergence
Continuous
Finite Power (energy over a period)
Dirichlet conditions:I Absolutely integrableI Bounded VariationI Finite Discontinuities
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 13 / 24
Linearity
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 14 / 24
Time-shift
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 15 / 24
Time Reversal
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 16 / 24
Time Scaling
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 17 / 24
Multiplication
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 18 / 24
Conjugate
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 19 / 24
Parseval’s Theorem
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 20 / 24
Discrete Time
Aliasing:
All periodic exponential signals with period N are:
φk [n] = e jk2πNn for k = 0, 1, ...,N − 1.
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 21 / 24
Discrete Time Fourier Series
x [n] =∑
k=<N>
akφk [n]
ak =1
N
∑n=<N>
x [n]φk [−n]
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 22 / 24
Multiplication
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 23 / 24
Fourier Series Example
Fourier Series Example using Matlab
x(t) = e−t for − 1 < t ≤ 1.
Cuff (Lecture 5) ELE 301: Signals and Systems Fall 2011-12 24 / 24
