The document discusses triangular wave functions and half-wave rectified sinusoidal functions. It defines a triangular wave function with period 2a and amplitude Xo that is shifted down by -Xo/2. It also defines a half-wave rectified sinusoidal function as the positive part of a sinusoid with period π, equal to sin(t) from 0 to π and 0 from π to 2π. The Fourier series representations of these functions are presented.

1. Triangular Wave Function
• The triangular wave function with period 2a is define by
f(t) = t = 2a-t 0 < t < a
a < t < 2a
F(t)
a0 2a
a
3a 4a

2. • Express the Fourier Series
for a triangular waveform?
• Express the Fourier Series
for a triangular waveform
that is amplitude shifted
down by –X0/2 ? Plot the
signal.
To
Xo
To
Xo/2
-Xo/2

3. • Express the Fourier Series for a
triangular waveform?
• Express the Fourier Series for a
triangular waveform that is
amplitude shifted down by –X0/2
? Plot the signal.
 
tkj
oddk
oo o
e
k
XX
tx 






/
2
2
2
)(
To
Xo
To
Xo/2
-Xo/2

4. HALF WAVE RECTIFIED SINUSOIDAL
FUNCTION

5. What Is “HALF WAVE SINE FUNCTION?”
A Sine wave or Sinusoid is a curve that
describes smooth repetitive oscillation.
Half Wave Sine Function is the representation
of positive part of the sine function.

6. HALF WAVE RECTIFIED SINUSOIDAL
FUNCTION...
It has a period of 2∏(Pi).
It is defined as a function of f(x) with period
2𝜋,
•F(x) = { sinx 0<x<𝜋
=0 𝜋 <x<2 𝜋 }
•F(x+2𝜋) =f(x)

7. The triangular wave rectified sinusoidal function with period 𝜋
is define by
f(t) = a sin t 0 < t < 𝜋
F(t)
a
0 𝜋 2𝜋
Full Wave Rectified Sinusoidal Function

8. ,sin)(
,sin)(
txf
txf




.0
,0


t
t


Since f(t) defined here is even, no terms of the form sin𝜔t will appear.
,
4
)(sin
2
)(sin
1
)(sin
1
0
0
0
0














ttd
ttdttda
F(x)
𝜋-2𝜋 0 𝜋 2𝜋

9. is not an orthogonality interval for both sines and cosines
.0
,
1
22
)(cossin
2
2
0
oddn
evenn
n
ttdntan




 




Note carefully that  ,0
together and we do not get zero for even n. The resulting series is
.
1
cos42
)(
,6,4,2
2

 

n n
tn
tf

