4.10 Frequency domain problems  (Page 4/6)

Circuits filter!

A unit-amplitude pulse with duration of one second serves as the input to an RC-circuit having transfer function $$H(f)=\frac{i\times 2\pi f}{4+i\times 2\pi f}$$

1. How would you categorize this transfer function: lowpass, highpass, bandpass, other?
2. Find a circuit that corresponds to this transfer function.
3. Find an expression for the filter's output.

Reverberation

Reverberation corresponds to adding to a signal its delayed version.

1. Assuming $\tau$ represents the delay, what is the input-output relation for a reverberation system? Is the systemlinear and time-invariant? If so, find the transfer function; if not, what linearity or time-invariancecriterion does reverberation violate.
2. A music group known as the ROwls is having trouble selling its recordings. The record company'sengineer gets the idea of applying different delay to the low and high frequencies and adding theresult to create a new musical effect. Thus, the ROwls' audio would be separated into two parts (oneless than the frequency $f_0$ , the other greater than $f_0$ ), these would be delayed by ${\tau }_l$ and ${\tau }_h$ respectively, and the resulting signals added. Drawa block diagram for this new audio processing system, showing its various components.
3. How does the magnitude of the system's transfer function depend on the two delays?

Echoes in telephone systems

A frequently encountered problem in telephones is echo. Here, because of acoustic coupling between the ear pieceand microphone in the handset, what you hear is also sent to the person talking. That person thus not onlyhears you, but also hears her own speech delayed (because of propagation delay over the telephonenetwork) and attenuated (the acoustic coupling gain is less than one). Furthermore, the same problem appliesto you as well: The acoustic coupling occurs in her handset as well as yours.

1. Develop a block diagram that describes this situation.
2. Find the transfer function between your voice and what the listener hears.
3. Each telephone contains a system for reducing echoes using electrical means. What simple system couldnull the echoes?

Effective drug delivery

In most patients, it takes time for the concentration of an administered drug to achieve a constant level in the blood stream. Typically, if the drug concentration in the patient's intravenous line is $C_{d}u(t)$ , the concentration in the patient's blood stream is $C_p(1-e^{-(at)})u(t)$ .

1. Assuming the relationship between drug concentration in the patient's drug and the delivered concentration can be described as a linear, time-invariant system, what is the transfer function?
2. Sometimes, the drug delivery system goes awry and delivers drugs with little control. What would the patient's drug concentration be if the delivered concentration were a ramp?More precisely, if it were $C_{d}tu(t)$ ?
3. A clever doctor wants to have the flexibility to slow down or speed up the patient's drug concentration. In other words, the concentration is to be $C_p(1-e^{-(bt)})u(t)$ , with $b$ bigger or smaller than $a$ . How should the delivered drug concentration signal be changed to achieve this concentration profile?