Debdut
Full Member level 3
I have a signal cos(w0 * t). It is sampled at frequency f_s. The samples are applied to a moving average filter of length L. I want to find the amplitude of the output.
So far I understand the following. Please do correct me if I am wrong; I am still learning the basics.
DTFT of impulse response of the moving average filter: H(w) = (1/L) * (1 - exp(-jwL)) / (1 - exp(-jw))
DTFT of input sampled sine wave: X(w) = pi * (Delta(w - w0) + Delta(w + w0))
DTFT of output: X(w)H(w) = (pi / L) * [ { (1 - exp(-jw0L)) / (1 - exp(-jw0)) * Delta(w - w0) } + { (1 - exp(jw0L)) / (1 - exp(jw0)) * Delta(w + w0) } ]
Now is the amplitude of output wave in time domain, say at w0, sqrt( (1 - cos w0L )^2 + (sin w0L)^2 ) / sqrt( (1 - cos w0 )^2 + (sin w0)^2 )
But the arguments of the sinusoids are having dimensions Hz, actually it should be unitless. Should I multiply the arguments by 1/( 2 * pi * f_s)?
So far I understand the following. Please do correct me if I am wrong; I am still learning the basics.
DTFT of impulse response of the moving average filter: H(w) = (1/L) * (1 - exp(-jwL)) / (1 - exp(-jw))
DTFT of input sampled sine wave: X(w) = pi * (Delta(w - w0) + Delta(w + w0))
DTFT of output: X(w)H(w) = (pi / L) * [ { (1 - exp(-jw0L)) / (1 - exp(-jw0)) * Delta(w - w0) } + { (1 - exp(jw0L)) / (1 - exp(jw0)) * Delta(w + w0) } ]
Now is the amplitude of output wave in time domain, say at w0, sqrt( (1 - cos w0L )^2 + (sin w0L)^2 ) / sqrt( (1 - cos w0 )^2 + (sin w0)^2 )
But the arguments of the sinusoids are having dimensions Hz, actually it should be unitless. Should I multiply the arguments by 1/( 2 * pi * f_s)?