Lecture 29 Review: Frequency response Frequency response examples Frequency response plots & signal spectra Filters Related educational materials: –Chapter

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. v in (t) is the input and v out (t) is the output

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. v S (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

RL circuit frequency response plots

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:

Plots of signal spectra Signal spectra plotted like frequency responses Amplitude and phase vs. frequency For our previous example:

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 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:

Annotate previous slide to calculate maximum value and frequency where we have times maximum value

RL circuit cutoff frequency Magnitude response:

Annotate to show calculation of cutoff frequency