Lowpass filter impulse response in frequency domain

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A Linear Time-Invariant system with impulse response h1[n]h1[n] is an ideal lowpass filter with cutoff frequency ωc=π/2ωc=π/2. The frequency response of the system is H1(ejω)H1(ejω). Suppose a new LTI system with impulse response h2[n]h2[n] is obtained from h1[n]h1[n] by h2[n]=(−1)n⋅h1[n]h2[n]=(−1)n⋅h1[n]. Sketch the frequency response H2(ejω)H2(ejω).
 

h2[n] is the product of two sequencies: h1[n] and (−1)^n .
Using convolution theorem you get H2(exp(jω)).
You can use shift theorem as well, noting that −1=exp(jΠ) , i.e. (−1)^n=exp(jnΠ) .

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