# Signalsandsystems

### Swamped with your writing assignments? We'll take the academic weight off your shoulders. We complete all our papers from scratch. You can get a plagiarism report upon request just to confirm.

Order a Similar Paper Order a Different Paper

CAN ANY OF YOU DO IT ? PLEASE REPLY ASAP.

Practical Assignment

Answer in no more than 6 pages total

Minimum 10pt font size

October 9, 2014

1. (Loopback test) Obtain an audio cable with 3.5mm stereo jacks on either end. Connect one end to

the audio output of your soundcard, and the other end to the audio input. Using the software available

on the course website (see folder called “loopback”), or your own software, play a sinusoidal signal of

frequency 100 Hz for a finite duration of time (atleast 2 seconds). Assert that you can hear the tone

when the cable is not plugged in and the audio is played through internal computer speakers, or a set of

connected headphones. Obtain samples at rate Fs = 44 100 Hz,

x1, . . . ,xL, y1, . . . ,yL

from the left and right channels of the soundcard input, where L is the number of samples obtained.

Reconstruct the signals as

x̃(t) =

L∑

`=1

x` sinc(Fst− `), ỹ(t) =

L∑

`=1

y` sinc(Fst− `)

where

sinc(t) =

sin(πt)

πt

. (1)

Plot these reconstructed signals for a 20 ms window from t = 1 s to 1.02 s.

Using scissors or wire cutters cut one of the channels of your audio cable. Alternatively, cut both and

reconnect one. Run the test again and plot the reconstructed signals for a 20 ms window from t = 1 s to

1.02 s.

2. (Multiplier) Consider the operational amplifier circuit in Figure 1. Analyse this circuit to obtain a

relationship between the input voltage x and the output voltage y. Build the circuit on a breadboard

and, using the soundcard, input the signal

x(t) = 1

3

sin(2πf1t) +

1

3

sin(2πf2t)

with f1 = 100 and f2 = 233. Using the (stereo) soundcard simultaneously record the input signal x

directly from the soundcard ouput and also the output voltage signal y. Build reconstructed approximate

signals x̃ and ỹ from the samples obtained and hypothesise a relationship between x̃ and ỹ. Plot x̃, ỹ and

the hypothesised signal over a 20 ms duration and comment on the validity of your hypothesis. List the

components that you used in constructing the circuit.

3. (Band-pass filter) Consider the operational amplifier circuit in Figure 2. Assuming the operational

amplifier is ideal, find a differential equation relating the input voltage signal x with the output voltage

signal y. Find the transfer function of the system H mapping x to y. Find the poles and zeros of the

system and construct a pole-zero plot. Assert that H is stable and regular and find its impulse response

h.

Build the circuit on a breadboard and, using a computer soundcard, input the signal

x(t) = 1

3

sin(2πf1t) +

1

3

sin(2πf2t)

with f1 = 500 and f2 = 1333. Using the (stereo) soundcard simultaneously record the input signal x

and also the output voltage signal y. Build reconstructed approximate signals x̃ and ỹ from the samples

obtained. Plot x̃, ỹ and H(x̃) = h∗ x̃ over a 4 ms duration. Assert that h∗ x̃ is close to ỹ. To compute

h∗ x̃ you may wish to use the trapezoidal integration method used in Test 5 of the lecture notes.

1

Now measure the spectrum of the circuit by using the soundcard to input sinusoidal signals of the form

xk(t) = sin(2πfkt), fk = 110 × 2k/2, k = 0, 1, . . . , 12.

For each k = 0, 1, . . . , 12 obtain an estimate of the spectrum Λ(H,fk). You may wish to use the method

described in Test 4 of the lecture notes. Find an analytical expression for the spectrum of the system

Λ(H,f) and plot the magnitude and phase spectrum over the interval f ∈ [0, 7500]. On the same plots

draw the measurements of the magnitude and phase spectrum obtained using the sound card. Assert

that the measurements conform with the hypothesised spectrum Λ(H,f). List the components used in

constructing the circuit.

4. (Butterworth filter) Design a lowpass second order Butterworth filter with cuttoff frequency in the

range 1800 Hz to 2200 Hz. Draw a diagram of the electrical circuit you have designed and list the compo-

nents. Derive the transfer function and the spectrum of your filter. Construct the circuit and, using the

computer soundcard, measure its spectrum over frequencies in the range 100 Hz to 7000 Hz. Plot your

measurements alongside the hypothesised spectrum that you derived.

2

−

+

x(t)

12 kΩ

y(t)22 kΩ

Figure 1: Operational amplifier circuit configured as a multiplier

−

+

3300 Ω

i(t)

100 nF

2200 Ω

x(t)

y(t)

10 nF

Figure 2: Operational amplifier configured as a band-pass filter with two capacitors and two resistors.

3

### We offer CUSTOM-WRITTEN, CONFIDENTIAL, ORIGINAL, and PRIVATE writing services. Kindly click on the ORDER NOW button to receive an A++ paper from our masters- and PhD writers.

Get a 10% discount on your order using the following coupon code SAVE10

Order a Similar Paper Order a Different Paper