7.1
Aperiodic (Nonperiodic) Signals
Finite duration signals (pulses) and random signals do not have repeating patterns.
These types of signals appear in many engineering applications. In order to develop
a frequency domain analysis for these signals, the Fourier Series will be applied,
where the fundamental period is considered to approach infinity.
Example:
Consider a train of square pulses (symmetric about t=0) with unit amplitude and
duration τ and period T0. Find the exponential Fourier Series in terms of τ and T0.
Plot the magnitudes of the Fourier coefficients as T0 approaches ∞ and observe what
is happening.
Show:
⎛ nπ ⎞
sin⎜ τ ⎟
⎝T ⎠
τ
D =
for n ≠ 0
n
π
T
τ
T
0
n
0
0
D =
0
τ
T
0
7.2
Fourier Coefficients for T0=5, tau=.5
Fourier Coefficients for T0=20, tau=.5
0.12
0.025
0.1
0.02
0.08
0.015
0.06
0.01
0.04
0.005
0.02
0
0
-0.005
-0.02
-2
-1.5
-1
-0.5
0
Hertz
0.5
1
1.5
2
-0.01
-2
Fourier Coefficients for T0=10, tau=.5
0.05
5
0.04
4
0.03
3
0.02
2
0.01
1
0
0
-0.01
-1
-0.02
-2
-1.5
-1
-0.5
0
Hertz
0.5
1
1.5
2
x 10
-2
-2
-1.5
-3
-1
-0.5
0
Hertz
0.5
1
1.5
2
1.5
2
Fourier Coefficients for T0=inf, tau=.5
-1.5
-1
-0.5
0
Hertz
0.5
1
7.3
The Fourier Transform:
Model an aperiodic signal, f(t), as a periodic signal with a period approaching
infinity. Write the Fourier series coefficients as a function of nω0 :
F ( nω ) 1
D =
=
∫ f ( t ) exp( − jnω t ) dt
T
T
The Fourier series expansion is now written as:
F ( nω )
f (t ) = ∑
exp( jnω t )
T
2π
= ω and substitute in above expressions to obtain:
Let Δω =
T
1
f (t ) =
∑ F ( nΔω ) Δω exp( jnΔωt )
2π
T0 / 2
0
n
0
0
0
− T0 / 2
∞
0
0
n =−∞
0
0
0
∞
n =−∞
T0 / 2
F ( nΔω ) = ∫ f ( t ) exp( − jnΔωt ) dt
− T0 / 2
Now let Δω → 0 ( T → ∞ ) (note that nΔω approaches a continuous variable)
0
7.4
The Fourier Transform:
F (ω ) = ∫ f ( t ) exp( − jωt ) dt
∞
−∞
The Inverse Fourier Transform:
1
f (t ) =
∫ F (ω ) exp( jtω)dω
2π
∞
−∞
Note the similarities between the Fourier Transform and the Laplace Transform.
Examples:
Find and plot the Fourier Transform of a Square Pulse p(t-t0)= u(t-t0+τ/2)-u(t-t0-τ/2).
ω
τ
Show: F (ω ) = τ exp( − jtω)
0
sin⎛⎜ ⎞⎟
⎝ 2⎠
ω
τ
2
Plot for τ=.1 and t0 = 2
ω
τ
= τ exp( − jtω)sinc⎛⎜ ⎞⎟
⎝ 2⎠
0
7.5
Magnitude of Delayed Rectangular P
0.1
0.09
0.08
Amplitude
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
-40
-30
-20
-10
0
Hz
10
P has e of Delay ed Rec tangular P uls e S pec trum
4
3
2
Phase in Radians
>>f = [-40:.05:40]; % Define frequency axis
>>w = 2*pi*f;
>>tau = .1;
% Define pulse parameter (width)
>>t0 = .2;
% Define pulse parameter (delay)
>>% Evaluate spectrum
>>spec = tau*exp(-j*w*t0).*sinc1(w, (2*pi)/tau);
>>
>>% Plot magnitude
>>figure(1)
>>plot(f, abs(spec), 'w');
>>title('Magnitude of Delayed Rectangular Pulse
Spectrum')
>>ylabel('Amplitude')
>>xlabel('Hz');
>>% Plot phase
>>figure(2)
>>plot(f, angle(spec), 'w')
>>title('Phase of Delayed Rectangular Pulse
Spectrum')
>>ylabel('Phase in Radians')
>>xlabel('Hz')
1
0
-1
-2
-3
-4
-40
-30
-20
-10
0
Hz
10
20
30
40
7.6
Example: Find inverse Fourier Transform of F (ω ) = π [ δ (ω − ω ) + δ (ω + ω )]
Show: f ( t ) = cos(ω t )
0
0
Example: Find Fourier Transform of f ( t ) = exp( − at )u ( t )
1
Show: F (ω ) =
for a > 0
a + jω
⎧⎪1 for t > 0
Example: Find Fourier Transform of f ( t ) = signum( t ) = ⎨
⎪⎩−1 for t < 0
2
Show: F (ω ) =
jω
0