Supplement: Instantaneous
frequency
Instantaneous Frequency
{
Treat cos(2𝜋𝑓! 𝑡)
Angle (degree)
Phase (ratio)
as a projection of 𝑒 #$%&! '
onto the x-axis.
1 d(2𝜋𝑓! 𝑡)
𝑓" 𝑡 =
= 𝑓!
2𝜋
d𝑡
2𝜋𝑓! 𝑡
cos(2𝜋𝑓! 𝑡)
Example. 𝑓! = 10 Hz
10 circulations per second
© Po-Ning Chen@ece.nctu
I-2
Instantaneous Frequency
{
Treat cos 2𝜋𝜙 𝑡
𝑓" 𝑡$ = 22 Hz
𝑓" 𝑡# = 10 Hz
Phase (ratio)
as a projection of 𝑒 #$%)(')
onto the x-axis
2𝜋𝜙(𝑡)
cos(2𝜋𝜙(𝑡))
1 d(2𝜋𝜙(𝑡))
𝑓" 𝑡 =
= 𝜙 ( (𝑡)
2𝜋
d𝑡
© Po-Ning Chen@ece.nctu
I-3
By this interpretation,
cos(2𝜋𝜙(𝑡)) and
𝑟 𝑡 cos(2𝜋𝜙(𝑡)) should
have the same
instantaneous frequency.
𝑓" 𝑡# = 10 Hz
𝑓" 𝑡$ = 22 Hz
2𝜋𝜙(𝑡)
cos(2𝜋𝜙(𝑡))
1 d(2𝜋𝜙(𝑡))
𝑓" 𝑡 =
= 𝜙 % (𝑡)
2𝜋
d𝑡
This explains why we add a
limiter in the FM
demodulation process.
FM
signal s(t)
Bandpass
filter
Noise w(t)
© Po-Ning Chen@ece.nctu
x(t)
Limiter
Discriminator
v(t) Baseband
lowpass
filter
Output
signal
Remove 𝑟(𝑡)
I-4
By this interpretation,
cos(2𝜋𝜙(𝑡)) and
𝑒 #$%)(') should have the
same instantaneous
frequency.
1 d(2𝜋𝜙(𝑡))
𝑓" 𝑡 =
= 𝜙 % (𝑡)
2𝜋
d𝑡
© Po-Ning Chen@ece.nctu
𝑓" 𝑡# = 10 Hz
𝑓" 𝑡$ = 22 Hz
2𝜋𝜙(𝑡)
cos(2𝜋𝜙(𝑡))
I-5
Discussion (Personal View)
Question: What is the
instantaneous frequency of
cos 2𝜋𝑡 + cos(4𝜋𝑡) ?
𝑓" 𝑡
4𝜋𝑡
cos(4𝜋𝑡)
2𝜋𝑡
cos(2𝜋𝑡)
No universally good answer.
© Po-Ning Chen@ece.nctu
I-6
Discussion (Personal View)
Same Question: What is the
instantaneous frequency of
𝑒 #$%' +𝑒 #,%' ?
𝑓" 𝑡
4𝜋𝑡
2𝜋𝑡
cos(2𝜋𝑡)
© Po-Ning Chen@ece.nctu
I-7
Derivation of Doppler Shift
© Po-Ning Chen@ece.nctu
I-8
© Po-Ning Chen@ece.nctu
I-9