I. Previously on IET
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Transcript of I. Previously on IET
I. Previously on IET
© Tallal Elshabrawy 2
Complex Exponential Function
( ) ( )tωsinj+tωcos=e tωj
Re-Axis
Im-Axis
ω
© Tallal Elshabrawy 3
Complex Exponential Function
( ) ( )tωsinj+tωcos=e tωj
Re-Axis
Im-Axis
ω
© Tallal Elshabrawy 4
Complex Exponential Function
( ) ( )tωsinj+tωcos=e tωj
Re-Axis
Im-Axis
ω
© Tallal Elshabrawy 5
Complex Exponential Function
( ) ( )tωsinj+tωcos=e tωj
Re-Axis
Im-Axis
ω
© Tallal Elshabrawy 6
Complex Exponential Function
( ) ( )tωsinj+tωcos=e tωj
Re-Axis
Im-Axis
ω
© Tallal Elshabrawy 7
Complex Exponential Function
( ) ( )tωsinj+tωcos=e tωj
Re-Axis
Im-Axis
ω
© Tallal Elshabrawy 8
Complex Exponential Function
( ) ( )tωsinj+tωcos=e tωj
Re-Axis
Im-Axis
ω
© Tallal Elshabrawy 9
Complex Exponential Function
( ) ( )tωsinj+tωcos=e tωj
Re-Axis
Im-Axis
ω
© Tallal Elshabrawy 10
Complex Exponential Function
( ) ( )tωsinj+tωcos=e tωj
Re-Axis
Im-Axis
ω
© Tallal Elshabrawy 11
Complex Exponential Function
( ) ( )tωsinj+tωcos=e tωj
Re-Axis
Im-Axis
ω
© Tallal Elshabrawy 12
The Fourier TransformRepresenting functions in terms of complex exponentials with different frequencies
jωt j2πftG f g t e dt g t e dt
© Tallal Elshabrawy 13
The Fourier Transform (Cosine Function)
Re-Axis
Im-Axis
-ωRe-Axis
Im-Axis
ω +
jωte cos ωt jsin ωt jωte cos ωt jsin ωt
jωt jωte e 2cos ωt
+
© Tallal Elshabrawy 14
The Fourier Transform (Cosine Function)
Re-Axis
Im-Axis
Re-Axis
Im-Axis
+
+
ω-ω
jωte cos ωt jsin ωt jωte cos ωt jsin ωt
jωt jωte e 2cos ωt
© Tallal Elshabrawy 15
The Fourier Transform (Cosine Function)
Re-Axis
Im-Axis
Re-Axis
Im-Axis
+
+
ω-ω
jωte cos ωt jsin ωt jωte cos ωt jsin ωt
jωt jωte e 2cos ωt
© Tallal Elshabrawy 16
The Fourier Transform (Cosine Function)
Re-Axis
Im-Axis
Re-Axis
Im-Axis
+
+
ω-ω
jωte cos ωt jsin ωt jωte cos ωt jsin ωt
jωt jωte e 2cos ωt
© Tallal Elshabrawy 17
The Fourier Transform (Cosine Function)
Re-Axis
Im-Axis
Re-Axis
Im-Axis
+
+
ω-ω
jωte cos ωt jsin ωt jωte cos ωt jsin ωt
jωt jωte e 2cos ωt
© Tallal Elshabrawy 18
The Fourier Transform (Sine Function)
Re-Axis
Im-Axis
Re-Axis
Im-Axis
- jωt jωte e 2jsin ωt
jωte cos ωt jsin ωt jωte cos ωt jsin ωt
ω-ω
-
© Tallal Elshabrawy 19
The Fourier Transform (Sine Function)
Re-Axis
Im-Axis
Re-Axis
Im-Axis
+ jωt jωte e 2jsin ωt
jωte cos ωt jsin ωt
ω-ω
+
jωte cos ωt jsin ωt
© Tallal Elshabrawy 20
The Fourier Transform (Sine Function)
Re-Axis
Im-Axis
Re-Axis
Im-Axis
- jωt jωtj e e 2sin ωt
jωtje jcos ωt sin ωt jωtje jcos ωt sin ωt
ω -ω
-
© Tallal Elshabrawy 21
The Fourier Transform (Sine Function)
Re-AxisRe-Axis
Im-Axis
+ jωt jωtj e e 2sin ωt
jωtje jcos ωt sin ωt jωtje jcos ωt sin ωt
ω
-ω
+
Im-Axis
© Tallal Elshabrawy 22
Fourier Transform of Sinusoids
Notes A real value for the coefficients in the frequency domain means that the starting
point for rotation is on the real axis An Imaginary value for the coefficients in the frequency domain means that the
starting point for rotation is on the imaginary axis
jωt jωte ecos ωt
2
jωt jωte e
sin ωt j2
0 ω-ω
1/21/2
0 ω-ω-j(1/2)
j(1/2)
© Tallal Elshabrawy 23
Fourier Transform of Real Valued Functions
A real-valued function in time implies thatG(-f) = G*(f)
Re-Axis
Im-Axisω1
Re-Axis
Im-Axis
ω2
Re-Axis
Im-Axisωn
Re-Axis
Im-Axis
-ω1
Re-Axis
Im-Axis-ω2
Re-Axis
Im-Axis
-ωn