dealing with aliasing
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From: fisico32 on 18 Jun 2010 16:10 Hello Forum, if a continuous-time sinusoid x(t)=cos(2pi*f1*t+theta) is sampled at an arbitrary rate f_s, we will obtain a sequence x[n]. This sequence will be the same sequence we would obtain from sampling, at the same rate f_s, an infinite number of other sinusoids of continuous frequency different from f1. The spectrum would be a series of delta at locations f =f1 +- k*f_s. This would happens even if f_s>>2*f1: the ambiguity on which continuous frequency sinusoid this sequence x[n] belongs to will remain... Does that mean that we need to make some assumptions, that we need to know something beforehand, to be able to associate x[n] to cos(2pi*f1*t+theta)? What are these assumptions? Aliasing occurs if our interpolating algorithm assumes that f_max=f_s/2, correct? If f1=5 Hz f_s/2=8 Hz, an ideal passband filter (-f_s/2 to f_s/2) in the freq. domain will pass f1=5 but also the frequency f=3 Hz, correct? thanks fisico32
From: Rune Allnor on 18 Jun 2010 16:36 On 18 Jun, 22:10, "fisico32" <marcoscipioni1(a)n_o_s_p_a_m.gmail.com> wrote: > Hello Forum, > > if a continuous-time sinusoid x(t)=cos(2pi*f1*t+theta) is sampled at an > arbitrary rate f_s, we will obtain a sequence x[n]. > > This sequence will be the same sequence we would obtain from sampling, at > the same rate f_s, an infinite number of other sinusoids of continuous > frequency different from f1. > The spectrum would be a series of delta at locations f =f1 +- k*f_s. > > This would happens even if f_s>>2*f1: the ambiguity on which continuous > frequency sinusoid this sequence x[n] belongs to will remain... > > Does that mean that we need to make some assumptions, No. > that we need to know > something beforehand, to be able to associate x[n] to cos(2pi*f1*t+theta)? Yes. > What are these assumptions? There aren't any. Rune
From: Tim Wescott on 18 Jun 2010 16:42
On 06/18/2010 01:10 PM, fisico32 wrote: > Hello Forum, > > if a continuous-time sinusoid x(t)=cos(2pi*f1*t+theta) is sampled at an > arbitrary rate f_s, we will obtain a sequence x[n]. > > This sequence will be the same sequence we would obtain from sampling, at > the same rate f_s, an infinite number of other sinusoids of continuous > frequency different from f1. > The spectrum would be a series of delta at locations f =f1 +- k*f_s. > > This would happens even if f_s>>2*f1: the ambiguity on which continuous > frequency sinusoid this sequence x[n] belongs to will remain... > > Does that mean that we need to make some assumptions, that we need to know > something beforehand, to be able to associate x[n] to cos(2pi*f1*t+theta)? > What are these assumptions? The obvious one: that f1 < f_s/2. Generally you enforce this with anti-aliasing filters. Note that you can also sample to a lower frequency, by sampling a high frequency bandpass signal and intentionally aliasing it down to DC. There's a term for it that escapes me at the moment... > Aliasing occurs if our interpolating algorithm assumes that f_max=f_s/2, > correct? Aliasing is a consequence of sampling, depending on how you view things it either occurs any time you sample, or whenever you sample a signal with frequency components outside of your assumed range. > If f1=5 Hz f_s/2=8 Hz, an ideal passband filter (-f_s/2 to f_s/2) > in the freq. domain will pass f1=5 but also the frequency f=3 Hz, correct? Yes. But if f_s = 16Hz, then the alias frequency for 5Hz will be 11Hz, not 3Hz. This may help: http://www.wescottdesign.com/articles/Sampling/sampling.html -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com |