Multi-Channel Aliasing Error for a Class of Almost Periodic Inputs

When such a system is presented with a signal % % MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaabm % aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqa % aiaaikdacqaHapaCaaWaa8qCaeaacaWGybWaaeWaaeaacqaHjpWDai % aawIcacaGLPaaacaWGLbWaaWbaaSqabeaacaWGPbGaeqyYdCNaamiD % aaaakiaadsgacqaHjpWDcaGGSaGaamiwamaabmaabaGaeqyYdChaca % GLOaGaayzkaaGaeyicI4SaamitamaaCaaaleqabaGaaGymaaaakmaa % bmaabaGaamOuaaGaayjkaiaawMcaaiabgMIihlaadYeadaahaaWcbe % qaaiaaikdaaaGcdaqadaqaaiaadkfaaiaawIcacaGLPaaaaSqaaiab % gkHiTiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aaaa!5FFE! $$x\left( t \right) = \frac{1}{{2\pi }}\int\limits_{ - \infty }^\infty {X\left( \omega \right){e^{i\omega t}}d\omega,X\left( \omega \right) \in {L^1}\left( R \right) \cap {L^2}\left( R \right)} $$ not necessarily bandlimited to |ω|<σ, the resulting error between the applied input and the multi-channel output, measured in the sup norm (aliasing error), is bounded by % % MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aa8 % quaeaadaabdaqaaiaadIfadaqadaqaaiabeM8a3bGaayjkaiaawMca % aaGaay5bSlaawIa7aiaadsgacqaHjpWDaSqaamaaemaabaGaeqyYdC % hacaGLhWUaayjcSdGaeyipaWJaeq4WdmhabeqdcqGHRiI8aaaa!4BB4! $$\mu \int\limits_{\left| \omega \right| < \sigma } {\left| {X\left( \omega \right)} \right|d\omega } $$ , [7], the constant of proportionality µ depending only on the system filters and independent of both σ and the particular input. Note that x(t) and X(ω) constitute a Fourier transform pair in the normalization shown. Here, we consider the same σ - designed multi-channel system and extend earlier results to almost periodic inputs of the form % % MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca % WGHbWaaSbaaSqaaiaad6gaaeqaaOGaamyzamaaCaaaleqabaGaamyA % aiabeU7aSnaaBaaameaacaWGUbaabeaaliaadshaaaaabaGaeyOeI0 % IaeyOhIukabaGaeyOhIukaniabggHiLdGccaGGSaWaaiWaaeaacqaH % 7oaBdaWgaaWcbaGaamOBaaqabaaakiaawUhacaGL9baaaaa!49AD! $$\sum\limits_{ - \infty }^\infty {{a_n}{e^{i{\lambda _n}t}}},\left\{ {{\lambda _n}} \right\}$$ real, % % MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada % abdaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakiaawEa7caGLiWoa % cqGH8aapcqGHEisPaSqaaiabgkHiTiabg6HiLcqaaiabg6HiLcqdcq % GHris5aaaa!43AA! $$\sum\limits_{ - \infty }^\infty {\left| {{a_n}} \right| < \infty } $$ and λn ≠ −σ + lc for integers n and % % MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeS4eHWMaai % ilaiaadogacqGH9aqpdaWcaaqaaiaaikdacqaHdpWCaeaacaWGTbaa % aaaa!3D43! $$\ell,c = \frac{{2\sigma }}{m}$$ . The resulting aliasing error, estimated using the sup norm, is shown to be bounded by % % MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdC2aaa % buaeaadaabdaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakiaawEa7 % caGLiWoaaSqaamaaemaabaGaeq4UdW2aaSbaaWqaaiaad6gaaeqaaa % WccaGLhWUaayjcSdGaeyOpa4Jaeq4WdmhabeqdcqGHris5aaaa!47BA! $$\gamma \sum\limits_{\left| {{\lambda _n}} \right| >\sigma } {\left| {{a_n}} \right|} $$ , where γ is a constant independent of σ and the input. In particular, the error is zero for such inputs when the {λn} satisfy |λn|<σ, λn≠−σ+lc for all n ∊ N and l =1, ⋯, m − 1. For the general case, the aliasing error approaches zero as the σ in the system design becomes infinite.