Hi all, In a manuals solution, I see when compute the discrete version with period T of z(t) where z(t)=x(t) * h(t), and they write: z(Tn)=x(Tn) * h(Tn). Where * is convolution, Based on this, can I refer that the following (1) and (2) are equal? 1. z(t)=x(t)*h(t) z(n)=z(nT). 2. x(n)=x(nT), h(n)=h(nT), z(n)=x(n)*h(n). I mean, is z(n) in (1) is equal to z(n) in (2)? Thanks
sampling question.
Started by ●August 8, 2007
Reply by ●August 8, 20072007-08-08
>Hi all, > >In a manuals solution, I see when compute the discrete version with >period T of z(t) where > >z(t)=x(t) * h(t), and they write: > >z(Tn)=x(Tn) * h(Tn). Where * is convolution, > >Based on this, can I refer that the following (1) and (2) are equal? > >1. z(t)=x(t)*h(t) > >z(n)=z(nT). > > >2. x(n)=x(nT), h(n)=h(nT), z(n)=x(n)*h(n). > >I mean, is z(n) in (1) is equal to z(n) in (2)? > >Thanks > >The following is from wikipedia (http://en.wikipedia.org/wiki/Sample_(signal)): Let z(t) be a continuous signal which is to be sampled, and that sampling is performed by measuring the value of the continuous signal every T seconds. Thus, the sampled signal z[n] is given by z[n] = z(nT) with n = 0,1,2,3,.... The sampling frequency or sampling rate fs is defined as the number of samples obtained in one second, or fs = 1 / T. The sampling rate is measured in hertz or in samples per second. Hope that helps! John
Reply by ●August 8, 20072007-08-08
On Aug 8, 4:20 pm, "A.E lover" <aelove...@gmail.com> wrote:> Hi all, > > In a manuals solution, I see when compute the discrete version with > period T of z(t) where > > z(t)=x(t) * h(t), and they write: > > z(Tn)=x(Tn) * h(Tn). Where * is convolution, > > Based on this, can I refer that the following (1) and (2) are equal? > > 1. z(t)=x(t)*h(t) > > z(n)=z(nT).z[n] = z(nT) the convention in many DSP texts (and a convention that i support without reservation, i think it should be the convention in all DSP papers and lit) is that arguments in brackets [n] are integers. so z[n] is a discrete sequence, not a continuous-time function. in the olden days they would write z n ("z sub n") but when we had a family of continuous-time signals in a multi-input or multi-output device (like you would get in state- variable control systems), then those signals already had subscripts and we needed a way of preserving notation so that z [n] 1 is the discrete-time version of z (t) 1 where t = n*T. r b-j
Reply by ●August 8, 20072007-08-08
Hi all, It seems that you guys still don't understand my question. Sorry for making your confusion. Let's notate (*) a convolution operator, * normal multiplying operator. z(t),x(t), and h(t) signals in continuous time domain, z[n],x[n],z[n] are signals in discrete time domain. I have z(t)=x(t) (*) h(t) all are in continuous time domain. Now my question is that if I sample z(t) by sampling rate Fs=1/T in order to obtain a discrete signal z[n] , can I write z(n*T)=x(n*T) (*) h (n*T) and therefore z[n]=x[n] (*) h[n] In other words, is sampling after convolution equal to sampling first and then taking convolution? Please help. Thanks On Aug 8, 4:42 pm, robert bristow-johnson <r...@audioimagination.com> wrote:> On Aug 8, 4:20 pm, "A.E lover" <aelove...@gmail.com> wrote: > > > Hi all, > > > In a manuals solution, I see when compute the discrete version with > > period T of z(t) where > > > z(t)=x(t) * h(t), and they write: > > > z(Tn)=x(Tn) * h(Tn). Where * is convolution, > > > Based on this, can I refer that the following (1) and (2) are equal? > > > 1. z(t)=x(t)*h(t) > > > z(n)=z(nT). > > z[n] = z(nT) > > the convention in many DSP texts (and a convention that i support > without reservation, i think it should be the convention in all DSP > papers and lit) is that arguments in brackets [n] are integers. so > z[n] is a discrete sequence, not a continuous-time function. in the > olden days they would write > > z > n > > ("z sub n") but when we had a family of continuous-time signals in a > multi-input or multi-output device (like you would get in state- > variable control systems), then those signals already had subscripts > and we needed a way of preserving notation so that > > z [n] > 1 > > is the discrete-time version of > > z (t) > 1 > > where t = n*T. > > r b-j
Reply by ●August 8, 20072007-08-08
"A.E lover" <aelover11@gmail.com> writes:> Hi all, > > It seems that you guys still don't understand my question. Sorry for > making your confusion. > Let's notate (*) a convolution operator, * normal multiplying > operator. z(t),x(t), and h(t) signals in continuous time domain, > z[n],x[n],z[n] are signals in discrete time domain. > > I have z(t)=x(t) (*) h(t) all are in continuous time domain. > > Now my question is that if I sample z(t) by sampling rate Fs=1/T in > order to obtain a discrete signal z[n] , can I write > > z(n*T)=x(n*T) (*) h (n*T) > > and therefore z[n]=x[n] (*) h[n] > > In other words, is sampling after convolution equal to sampling first > and then taking convolution?Yes, provided that your sample rate is high enough that neither x[n] nor h[n] are aliased. However, you are obfuscating the issue somewhat by using two different notations for the same discrete-time signal: x(n*T) and x[n]. They are equivalent. The "x[n]" (rather than "x(n*T)") notation is preferable, in my opinion. -- % Randy Yates % "She tells me that she likes me very much, %% Fuquay-Varina, NC % but when I try to touch, she makes it %%% 919-577-9882 % all too clear." %%%% <yates@ieee.org> % 'Yours Truly, 2095', *Time*, ELO http://home.earthlink.net/~yatescr
Reply by ●August 8, 20072007-08-08
Thanks Randy, however I doubt this statement. Let's assume X(0)=H(0)=1, where X(f), H(f) are the continuous fourier transform of x(t) and h(t). (x(t), h(t) are in continuous domain). I take z(t)=x(t)(*) h(t) first, so in frequency domain clearly Z(0)=X(0)*H(0)= 1*1=1. then I sample z(t) at Fs to obtain z[n]=z(n/Fs). In frequency domain of Z(w), a DTFT of z[n], we have Z(w=0)=Fs*1=Fs, according to the equation of relationship between (DTFT of z[n]) and (CTFT of z(t). Now consider the case I sample two continuous function x(t) and h(t) firstly to obtain x[n] and h[n], then convolute them to obtain z[n]=x[n](*)h[n]. Similarly, X(w=0)=Fs, H(w=0)=Fs,==> Z(w=0)=X(w=0)*H(w=0)= Fs^2. I see two ways are not equivalent in frequency domain. What do you think? or am I missing some points? Thanks> Yes, provided that your sample rate is high enough that neither x[n] nor > h[n] are aliased. > > However, you are obfuscating the issue somewhat by using two different > notations for the same discrete-time signal: x(n*T) and x[n]. They are > equivalent. The "x[n]" (rather than "x(n*T)") notation is preferable, > in my opinion. > -- > % Randy Yates % "She tells me that she likes me very much, > %% Fuquay-Varina, NC % but when I try to touch, she makes it > %%% 919-577-9882 % all too clear." > %%%% <ya...@ieee.org> % 'Yours Truly, 2095', *Time*, ELO http://home.earthlink.net/~yatescr
Reply by ●August 9, 20072007-08-09
"A.E lover" <aelover11@gmail.com> writes:> Thanks Randy, however I doubt this statement. > > Let's assume X(0)=H(0)=1, where X(f), H(f) are the continuous fourier > transform of x(t) and h(t). (x(t), h(t) are in continuous domain). > > I take z(t)=x(t)(*) h(t) first, > so in frequency domain clearly Z(0)=X(0)*H(0)= 1*1=1. > then I sample z(t) at Fs to obtain z[n]=z(n/Fs). > In frequency domain of Z(w), a DTFT of z[n], we have Z(w=0)=Fs*1=Fs, > according to the equation of relationship between (DTFT of z[n]) and > (CTFT of z(t). > > > Now consider the case I sample two continuous function x(t) and h(t) > firstly to obtain x[n] and h[n], then convolute them to obtain > z[n]=x[n](*)h[n]. > > Similarly, X(w=0)=Fs, H(w=0)=Fs,==> Z(w=0)=X(w=0)*H(w=0)= Fs^2. > > I see two ways are not equivalent in frequency domain. > > What do you think? or am I missing some points?Your thinking is correct and I am the one who made the error. Let xd[n] = x(n*Ts), where Ts = 1 / Fs and Fs is the sample rate. Let Xd(w) be the DTFT of xd[n] and let X(w) be the continuous-time Fourier transform of x(t), where w = 2 * pi * f. Then Xd(w) = Fs * X(w). This Fs is the factor I had omitted but which you correctly accounted for. This result follows from modeling sampling as a modulation of the input signal x(t) with the infinite impulse train. See, e.g., [signalsandsystems]. Therefore the two methods are equivalent EXCEPT for a scale factor of Fs, as you've already discovered. --Randy @BOOK{signalsandsystems, title = "{Signals and Systems}", author = "{Alan~V.~Oppenheim, Alan~S.~Willsky, with Ian~T.~Young}", publisher = "Prentice Hall", year = "1983"} -- % Randy Yates % "I met someone who looks alot like you, %% Fuquay-Varina, NC % she does the things you do, %%% 919-577-9882 % but she is an IBM." %%%% <yates@ieee.org> % 'Yours Truly, 2095', *Time*, ELO http://home.earthlink.net/~yatescr