Lecture 1 - Digilent Inc.
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Transcript Lecture 1 - Digilent Inc.
Lecture 29
•Review:
• Frequency response
•Frequency response examples
•Frequency response plots & signal spectra
•Filters
•Related educational materials:
–Chapter 11.1 - 11.3
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 ) tan1
• 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 ) tan1
RL circuit frequency response plots
H ( j )
2
1 ( 2 )2
H ( j ) 90 tan1 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