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Lecture 29
•Review:
• Frequency response
•Frequency response examples
•Frequency response plots & signal spectra
•Filters
•Related educational modules:
–Section 2.8.1, 2.8.2
Frequency Response
• Systems are characterized in the frequency domain,
by their frequency response, H(j)
• Magnitude response: the ratio of the output amplitude to
the input amplitude as a function of frequency
• Phase response: the difference between the output phase
and the input phase, as a function of frequency
Magnitude and phase responses
• Output:
Review: RC circuit frequency response
• Determine the magnitude and phase responses of the circuit
below. vin(t) is the input and vout(t) is the output
H ( j ) 
2
4
2
 2
H ( j )   tan1 
• Annotate previous slide to denote |H|=mag
resp, <H = phase resp.
Example: RL circuit frequency response
• Determine the magnitude and phase responses of the circuit
below. vS(t) is the input and v(t) is the output
Frequency response plots
• Frequency responses are often presented graphically
in the form of two plots:
• Magnitude response vs. frequency
• Phase response vs. frequency
RC circuit frequency response plots
H ( j ) 
2
4
2
 2
H ( j )   tan1 
RL circuit frequency response plots
H ( j ) 
2
1  ( 2 )2
H ( j )  90  tan1 2 
Signal spectra
• The frequency domain content of a signal is called
the spectrum of the signal
• Example:
v(t) = 3cos(t+20) + 7cos(2t-60)
• Spectrum:
 3 20 , ω  1 rad / sec

V ( j )  7  60 , ω  2 rad / sec

0 , otherwise
Plots of signal spectra
• Signal spectra plotted like frequency responses
• Amplitude and phase vs. frequency
• For our previous example:
V ( j )
V ( j )
7
20
2
, rad/sec
0
1
3
, rad/sec
0
0
1
2
3
-60
3
Graphical interpretation of system response
• Plots of the input spectrum and frequency response
can combine to provide an output spectrum plot
• Point-by-point multiplication of magnitude plots
• Point-by-point addition of phase plots
• Can provide conceptual insight into system behavior
Example – RL circuit response to example input
Frequency selective circuits and filters
• Circuits are often categorized by the general
“shape” of their magnitude response
• The response in some frequency ranges will be high
relative to the input; these frequencies are passed
• H(j) is “large” in these frequency ranges
• The response in some frequency ranges will be low
relative to the input; these frequencies are stopped
• H(j) is “small” in these frequency ranges
Filters
• Circuits which select certain frequency ranges to
pass and other frequency ranges to stop are often
called frequency selective circuits or filters
• Example: audio system graphic equalizer
• The range (or band) of frequencies that are passed
by the filter is called the passband
• The range (or band) of frequencies that are stopped
by the filter is called the stopband
Specific case I – Lowpass filters
• Lowpass filters pass low
frequencies and stop high
frequencies
• The boundary between the
two bands is the cutoff
frequency, c
• “Low” frequencies are less
than c, “high” frequencies
are greater than c
• On previous slide, note that IDEAL filters
absolutely remove all components outside the
passband.
• Also point out that these cannot be implemented
in the real world (turns out that they would need
to respond to the input before the input is
applied – they need to see into the future)
Specific case II – Highpass filters
• Highpass filters pass high
frequencies and stop low
frequencies
• The boundary between the
two bands is (still) called
the cutoff frequency, c
Additional filter categories
• Filters are often categorized by the order of the differential
equation governing the circuit
• e.g. First order filter, second order filter
• Filters can also be bandpass or bandstop
• A band of frequencies between two cutoff frequencies is
either passed or stopped
• Lowpass & highpass filters can be first or higher order
• Bandpass & bandstop filters must be at least second order
• We will only work with first order filters in this course
Filter example 1 – Lowpass filter
• RC circuit:
Filter example 2 – Highpass filter
• RL circuit:
Non-ideal first order filters
• Realizable filters do not have sharp transitions
between the passband and stopband
• So where is the cutoff frequency (c)?
• Define the cutoff frequency where the magnitude
• response is 1 2 times the maximum magnitude
• Why?
• The power is (generally) the square of the signal  the
cutoff frequency is where we have half of the maximum
power (it is sometimes called the half power point)
RC circuit cutoff frequency
• Magnitude response:
H ( j ) 
2
4  2
• Annotate previous slide to calculate maximum
value and frequency where we have 0.707
times maximum value
RL circuit cutoff frequency
• Magnitude response:
H ( j ) 
2
1  ( 2 )2
• Annotate to show calculation of cutoff
frequency