# Signalsandsystems

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.

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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.

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