4.10 Frequency domain problems  (Page 3/6)

Lowpass filtering a square wave

Let a square wave (period $T$ ) serve as the input to a first-order lowpass systemconstructed as a RC filter. We want to derive an expression for the time-domain response of the filter tothis input.

1. First, consider the response of the filter to a simple pulse, having unit amplitude and width $\frac{T}{2}$ . Derive an expression for the filter's output to thispulse.
2. Noting that the square wave is a superposition of a sequence of these pulses, what is the filter'sresponse to the square wave?
3. The nature of this response should change as the relation between the square wave's period and thefilter's cutoff frequency change. How long must the period be so that the response does not achieve a relatively constant value between transitions in the squarewave? What is the relation of the filter's cutoff frequency to the square wave's spectrum in thiscase?

Mathematics with circuits

Simple circuits can implement simple mathematical operations, such as integration and differentiation. Wewant to develop an active circuit (it contains an op-amp) having an output that is proportional to theintegral of its input. For example, you could use an integrator in a car to determine distance traveled fromthe speedometer.

1. What is the transfer function of an integrator?
2. Find an op-amp circuit so that its voltage output is proportional to the integral of its input for allsignals.

Where is that sound coming from?

We determine where sound is coming from because we have two ears and a brain. Sound travels at a relativelyslow speed and our brain uses the fact that sound will arrive at one ear before the other. As shown here , a sound coming from the right arrives at the left ear $\tau$ seconds after it arrives at the right ear.

Once the brain finds this propagation delay, it can determine the sound direction. In an attempt to modelwhat the brain might do, RU signal processors want to design an optimal system that delays each ear's signal by some amount then adds themtogether. ${\Delta }_l$ and ${\Delta }_r$ are the delays applied to the left and right signalsrespectively. The idea is to determine the delay values according to some criterion that is based on what ismeasured by the two ears.

1. What is the transfer function between the sound signal $s(t)$ and the processor output $y(t)$ ?
2. One way of determining the delay $\tau$ is to choose ${\Delta }_l$ and ${\Delta }_r$ to maximize the power in $y(t)$ . How are these maximum-power processing delaysrelated to $\tau$ ?

Arrangements of systems

Architecting a system of modular components meansarranging them in various configurations to achieve some overall input-output relation. For each of the following , determine the overall transfer function between $x(t)$ and $y(t)$ .

System a

System b

System c

The overall transfer function for the cascade (first depicted system) is particularly interesting. What doesit say about the effect of the ordering of linear, time-invariant systems in a cascade?

Filtering

Let the signal $s(t)=\frac{\sin (\pi t)}{\pi t}$ be the input to a linear, time-invariant filter having the transfer function shown below . Find the expression for $y(t)$ , the filter's output.