In section 10.6, we saw that a system’s frequency response provided a steady-state input-output relationship for a system, as a function of frequency. We could apply this frequency response to the phasor representation of the input signal in order to determine the system’s steady-state sinusoidal response – we simply evaluated the frequency response at the appropriate frequencies to determine the effect of the system on the input sinusoids. This approach had the potential for simplifying our analysis considerably, particularly for the case in which the input signal contained multiple sinusoids with different frequencies.

In Chapter 10, the signals we considered consisted only of individual sinusoids. It is more useful, however, in some ways to think in terms of the inputs and outputs of the system as functions of frequency, in the same way in which we considered the frequency response of the system to be a function of frequency in section 10.6. We can then perform our analysis of the system entirely in terms of the frequencies involved. This leads to the use of the system’s frequency response directly as a design and analysis tool. In many cases, this means that the actual time- domain behavior of the system or signal is of limited interest (or in some cases, not considered at all). Some examples of frequency domain analyses are:

1. Determining dominant sinusoidal frequency components in a measured signal. Complex signals can often be represented as a superposition of several sinusoidal components with different frequencies. Identifying sinusoidal components with large amplitudes (the so-called dominant frequencies) can help with many design problems. One application of this is in the area of combustion instability – combustion processes in rocket engines can become unstable due to a variety of reasons, any of which can result in catastrophic failure of the engine. The type of instability which occurs is generally linked to a particular frequency; identification of the frequency of the pressure oscillations associated with the combustion instability is generally the first step in determining the cause of the instability.
2. Designing systems to provide a desired frequency response. Audio components in stereo systems are generally designed to produce a desired frequency response. A graphic equalizer, for example, can be used to boost (or amplify) some frequency ranges and attenuate other frequency ranges. When adjusting the settings on an equalizer, you are essentially directly adjusting the system’s frequency response to provide a desired system response.

This chapter begins in section 11.1 with a brief review of frequency responses and an overview of the use of the frequency response in system analysis and design. In section 11.2, we discuss representation of signals in terms of their frequency content. At this time, we will also represent the frequency content of the input and output signals and the frequency response of the system in graphical format – this helps us visualize the frequency content of the signals and system. This leads us to think in terms of using a system to create a signal with a desired frequency content – this process is called filtering and is discussed in section 11.3. Using logarithmic scales to represent the signal and system frequency responses can – in many cases – simplify the analysis or design process; this format of presentation is called a Bode plot, and they are very briefly introduced in section 11.4. We will discuss Bode plots in more depth in later chapters.

It is important to keep in mind that, when we are performing frequency domain analyses, we are restricting our attention to the steady-state sinusoidal response of the system. Frequency domain design and analysis methods are so pervasive that they are often used to infer the system’s transient response and/or its response to non- sinusoidal signals, so it is sometimes possible to forget the origins and limitations of the original concepts!

• Use the frequency response of a system to determine the frequency domain response of a system to a given input
• State from memory the definition of signal spectrum
• Create plots of given signal spectra
• Plot a circuit’s magnitude and phase responses
• Check a circuit’s amplitude response at low and high frequencies against the expected physical behavior of the circuit
• Graphically represent a system’s frequency domain response from provided signal spectra plots and plots of the system’s frequency response
• Identify low pass and high pass filters
• Calculate a system’s cutoff frequency
• Determine the DC gain of an electrical circuit
• Write, from memory, the equation used to convert gains to decibel form
• Sketch straight-line amplitude approximations to Bode plots
• Sketch straight-line phase approximations to Bode plots

In section 10.6, we defined the frequency response $H(j \omega)$ of a system as a complex function of frequency which describes the relationship between the steady state sinusoidal response of a system and the corresponding sinusoidal input. Thus, if a sinusoidal input with some frequency $\omega_0$ is applied to a system with frequency response $H(j \omega)$, the amplitude of the output sinusoid is the input sinusoid’s amplitude multiplied by the magnitude response of the system, evaluated at the frequency $\omega_0$. The phase angle of the output sinusoid is the sum of the input sinusoid’s phase and the phase response of the system, evaluated at the frequency $\omega_0$. The overall idea is presented in block diagram form in Figure 11.1 below.

The true power of the frequency response is, however, if we consider both the system’s input and output phasors to be complex functions of frequency, in the same way that the frequency response is a complex function of frequency. In this case, the block diagram of Figure 11.1 can be represented as shown in Figure 11.2.

In Figure 11.2, the output is determined by multiplying the phasor representation of the input by the system’s frequency response. It is important to keep in mind that the arguments of this multiplication are complex functions of frequency – both the input and the frequency response at any frequency are complex numbers, so the output at any frequency is also a complex number. We typically use polar form to represent these complex numbers, so the amplitude of the output signal is the product of the amplitude of the input signal and the magnitude response of the system and the phase of the output signal is the sum of the phase of the input and the phase response of the system. Mathematically, these are expressed as:

$$|\underline{Y}(j \omega)|=|\underline{U}(j \omega)| \cdot |H (j \omega)| (Eq. 11.1) And $$\angle \underline{Y}(j \omega) = \angle \underline{U}(j \omega) + \angle H(j \omega) (Eq. 11.2)$$ We now present two examples of the process defined by equations (11.1) and (11.2) above. ---- ==== Example 11.1 ==== Determine the phasor representation for $v_{out}(t)$ in the circuit shown below as a function of frequency, if the input voltage is $v_{in}(t) = 3 \cos(2t+20^{\circ}) + 7 \cos(4t-60^{\circ})$. (Note: this problem is the same as that of Example 10.19 of chapter 10.6; the difference is primarily philosophical.)

The frequency response of this circuit, for arbitrary resistance and capacitance values, was determined in example 10.18 of chapter 10.6. For our specific resistor and capacitor values, this becomes:

$$H(j \omega) = \frac{1}{1+j \omega RC} = \frac{1}{1+j \omega(2 \Omega)(0.25F)} = \frac{1}{1+j \omega(0.5)} = \frac{2}{2+j \omega}$$

We can represent the input as a piecewise function of frequency:

$$\underline{V}_{in} = \begin{cases} 3 \angle 20^{\circ}, & \omega = 2 \text{rad/sec} \\ 7 \angle -60^{\circ}, & \omega = 4 \text{rad/sec} \\ 0, & \text{otherwise} \end{cases} $$

The input phasor is now considered to be a function of frequency, whose only nonzero components are at frequencies of 2 rad/sec and 4 rad/sec.

The phasor output is simply the product of the input phasor as a function of frequency and the frequency response. For frequencies other than 2 rad/sec and 4 rad/sec, the input is zero and the frequency response is finite, so the output is zero. We determined the output phasor at frequencies of 2 and 4 rad/sec in example 10.19 in section 10.6; using those results allows us to write the output phasor directly as:

$$\underline{V}_{out} = \begin{cases} \frac{3}{\sqrt{2}} \angle -25^{\circ}, & \omega = 2 \text{rad/sec} \\ \frac{7}{\sqrt{5}} \angle -123.4^{\circ}, & \omega = 4 \text{rad/sec} \\ 0, & \text{otherwise} \end{cases} $$

The frequency response of a system, $H(j \omega)$, and the frequency domain input to the system, $\underline{U}(j \omega)$, are given below. The frequency response is dimensionless, the input has units of volts, and the units of frequency are rad/sec. Determine the system output $\underline{Y}(j \omega)$.

$$\underline{U}_{in} = \begin{cases} 3 \angle 20^{\circ}, & \omega = 2 \text{rad/sec} \\ 7 \angle -60^{\circ}, & \omega = 4 \text{rad/sec} \\ 0, & \text{otherwise} \end{cases} $$