EE 435 Lecture 30 Data Converters Spectral Performance
. Review from last lecture. INL Often Not a Good Measure of Linearity Four identical INL with dramatically different linearity X OUT X OUT X REF X REF X IN X IN X REF X REF X OUT X OUT X REF X REF X IN X IN X REF X REF
. Review from last lecture. Linearity Issues INL is often not adequate for predicting the linearity performance of a data converter Distortion (or lac thereof) is of major concern in many applications Distortion is generally characterized in terms of the harmonics that may appear in a waveform
. Review from last lecture. Spectral Analysis If f(t) is periodic f(t) alternately f(t) A A T 0 A sin ω t θ 0 a s in t ω t b c o s ω ω π T A a b Termed the Fourier Series Representation of f(t)
. Review from last lecture. Distortion Analysis Total Harmonic Distortion, THD THD RMS voltage in harmonics RMS voltage of fundamenta l THD A A3 A A4... THD A A
Spurious Free Dynamic Range, SFDR The SFDR is the difference between the fundamental and the largest harmonic A SFDR 3 4 5 6 SFDR is usually determined by either the second or third harmonic
In a fully differential symmetric circuit, all even harmonics are absent in the differential output! A 3 4 5 6
Theorem: In a fully differential symmetric circuit, all even-order terms are absent in the Taylor s series output for symmetric differential excitations! Proof: V ID + - V O V O + V OD Expanding in a Taylor s series around V ID =0, we obtain - V f V h V 0 ID ID 0 V f -V h -V 0 ID ID 0 V =V V h V h -V OD 0 0 ID ID 0 0 V = h V -V OD ID ID 0 V = h V V OD ID ID 0 When is even, term in [ ] vanishes
Theorem: In a fully differential symmetric circuit, all even harmonics are absent in the differential output for symmetric differential excitations! V ID + V O + V OD Proof: - V O - Recall: sin n x n 0 n 0 h sin n x for nodd g sin n x for neven where h, g, and θ are constants That is, odd powers of sin n (x) have only odd harmonics present and even powers have only even harmonics present
Theorem: In a fully differential symmetric circuit, all even harmonics are absent in the differential output for symmetric differential sinusoidal excitations! V OD Proof: V ID + V f V h V - V O V O + V OD Expanding in a Taylor s series around V ID =0, we obtain Assume V ID =Ksin(ωt) W.L.O.G. assume K= V V h sin ω t V h -sin ω t O O and O ID ID 0 V 0 O 0 h O - O ID ID 0 ) V f -V h (-V 0 sin ω t -sin ω t h sin ω t sin ω t Observe the even-ordered powers and hence even harmonics are absent in this last sum 0
How are spectral components determined? By integral a or ωt t T A ωt t T t f t e jωt dt t T t f t e jωt dt f tsin tωdt b f tcos tωdt t ωt Integral is very time consuming, particularly if large number of components are required t T t By DFT (with some restrictions that will be discussed) By FFT (special computational method for obtaining DFT)
How are spectral components determined? T T S Consider sampling f(t) at uniformly spaced points in time T S seconds apart This gives a sequence of samples N f T s =
Consider a function f(t) that is periodic with period T f(t) A T 0 A sin ω t θ =f = T Band-limited Periodic Functions Definition: A periodic function of frequency f is band limited to a frequency f max if A =0 for all f max f
T NOTATION: T S T: Period of Excitation T S : Sampling Period N P : Number of periods over which samples are taen N: Total number of samples N P NT T S N h = Int - N P Note: N P is not an integer unless a specific relationship exists between N, T S and T Note: The function Int(x) is the integer part of x
T T S THEOREM (conceptual) : If a band-limited periodic signal is sampled at a rate that exceeds the Nyquist rate, then the Fourier Series coefficients can be directly obtained from the DFT of a sampled sequence. x T S N 0 Χ N 0
T A Χ m N 0 T S THEOREM: Consider a periodic signal with period T=/f and sampling period T S =/f S. If N P is an integer and x(t) is band limited to f MAX, then and 0 m h - Χ mn P for all not defined above where Χ N is the DFT of the sequence xt S 0 N=number of samples, N P is the number of periods, and N 0 fmax h = Int f N P Key Theorem central to Spectral Analysis that is widely used!!! and often abused
Why is this a Key Theorem? T A Χ m N 0 T S THEOREM: Consider a periodic signal with period T=/f and sampling period T S =/f S. If N P is an integer and x(t) is band limited to f MAX, then and where Χ N 0 m h - Χ mn P for all not defined above is the DFT of the sequence xt S 0 N=number of samples, N P is the number of periods, and N DFT requires dramatically less computation time than the integrals for obtaining Fourier Series coefficients Can easily determine the sampling rate (often termed the Nyquist rate) to satisfy the band limited part of the theorem 0 fmax h = Int f N P
How is this theorem abused? T A Χ m N 0 T S THEOREM: Consider a periodic signal with period T=/f and sampling period T S =/f S. If N P is an integer and x(t) is band limited to f MAX, then and where Χ N 0 m h - Χ mn P for all not defined above is the DFT of the sequence xt S 0 fmax h = Int f N Much evidence of engineers attempting to use the theorem when N P is not an integer Challenging to have N P an integer in practical applications Dramatic errors can result if there are not exactly an integer number of periods in the sampling window N=number of samples, N P is the number of periods, and N 0 P
T T S If the hypothesis of the theorem are satisfied, we thus have A A 0 A A 3 A4 N P + N P + 3N P + 4N P +
If the hypothesis of the theorem are satisfied, we thus have A A A 3 A4 A 0 N P + N P + 3N P + 4N P + FFT is a computationally efficient way of calculating the DFT, particularly when N is a power of
FFT Examples Recall the theorem that provided for the relationship between the DFT terms and the Fourier Series Coefficients required. The sampling window be an integral number of periods. The input signal is band limited to f MAX
End of Lecture 30