CORE1240 HW1

Q1

(a)

x_{1} = 5 sin (2 π 208 t) + sin (2 π 572 t) + 10 sin (2 π 780 t) GC D (208, 572, 780) = 52 T_{x_{1}} = \frac{1}{f _{x_{1}}} = \frac{1}{52} s x_{2} = 20 sin (2 π 392 t) + 5 sin (2 π 784 t) + 30 sin (2 π 1078 t) GC D (208, 572, 780) = 98 T_{x_{2}} = \frac{1}{f _{x_{2}}} = \frac{1}{98} s

(b)

No, they have different fundamental frequencies, so their pitch would be different.

Q2

(a)

s (t) = 7 sin (2 π 10000 t) + 4 sin (2 π 15000 t) + 5 sin (2 π 20000 t) + 3 sin (2 π 30000 t) + 2 sin (2 π 35000 t)

(b)

GC D (10000, 15000, 20000, 30000, 35000) = 5000 f_{s (t)} = 5000 Hz

(c)

The 3rd and 4th harmonics.

(d)

Band-pass filter.

(e)

At $15000 Hz$ , the filter’s amplitude is $1$ .
At $20000 Hz$ , the filter’s amplitude is $0.5$ .

HW1 Q2e

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Q3

(a)

The fundamental frequency of the signal is $h = 4 sin (2 π 1.25 t)$

HW1 Q3a

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(b)

The entire signal is $h = 4 sin (2 π 1.25 t) + 3 sin (2 π 2.5 t) + 2 sin (2 π 3.75 t) + 2 sin (2 π 5 t)$

HW1 Q3b

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