**Unformatted text preview: **oﬀ frequencies which satisfy (2.1) would be suﬃcient. Let’s say, for
practicality sake, that we want to choose the cutoﬀ frequencies in the midpoints between the
desired and undesired frequency components. This gives us the following speciﬁc values:
�1 = (0 + 1)�0
,
2 �2 = (2 + 3)�0
,
2 � �1 = ω
,
4 �2 = 7 �3 = (3 + 4)�0
,
2 5ω
,
4 �3 = 7ω
.
2 where �0 = ω
2 Problem 3 Consider a causal discrete-time LTI system whose input x[n] and output y [n]
are related by the following diﬀerence equation:
1
y [n] − y [n − 1] = x[n] + 2x[n − 4]
4
Find the Fourier series representation of the output y [n] when the input is
x[n] = 2 + sin(ωn/4) − 2 cos(ωn/2). First, let’s ﬁnd the frequency response of the system from the diﬀerence equation by injecting
an input, x[n], that is an eigenfunction of the LTI system:
x[n] = ej �n � y [n] = H (ej� ) ej�n
H (ej� ) is the frequency response characterizing the system or the eigenvalue of the system.
By substituting x[n] and y [n]...

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