MAE 2381: Chapters 6 & 7

Chapter 6: Windowing and Filtering - Chapter 7: Sampling

sampling rate

how often a sample is taken (a frequency, Hz)

f_s = 1/δt where:

f_s = sampling rate, Hz δt = time between samples, s

if sampling rate is too low, then the sampled signal can appear lower than the actual signal (aliasing)

nyquist frequency

the maximum frequency that can be detected for a given sampling rate

f_max = f_N = f_s/2 = 1/2δt where:

f_N = Nyquist frequency

minimum sample rate

to accurately reconstruct the frequency content of a measured signal

f_s >= 2f_max where:

f_s = minimum sample rate

f_max = maximum frequency of signal

maximum sample time increment

to accurately reconstruct the frequency content of a measured signal:

δt_s < 1/2f_max where:

δt_s = maximum sample time increment

alias frequency

a sampling rate too low causes frequencies to be misinterpreted → this false/lower frequency is called ____

folding diagram

a diagram in which the original input frequency axis is folded back over itself at the folding point of the Nyquist frequency and again for each of its harmonics

gain

ratio between the amplified amplitude (A_a) and the original amplitude (A₀)
G = A_a/A₀

magnitude ratio (M(w))

the degree of modification (ratio) when a signal’s amplitude is modified by some process (like filtering or a system response)

M(w) = M_m/M₀ where:

M_m = modified magnitude

M₀ = original magnitude

windowing

cleans up data in the time domain to reduce the effect of clipping

basically like zooming in on a section of data

clipping

incomplete sinusoid periods in sample windows

rectangular window

1 = keep the data 0 = throw it away

fixed interval: multiply desired data by 1 and the signals before/after it by 0

filtering

removes unwanted frequencies from a signal (in the frequency domain)

you have to convert everything from the time domain to the frequency domain to filter

• an analog electronic circuit can filter the signal before sampling

Hanning window

convert the data window into a periodic function that begins and ends at y = 0

• reduces the clipping effect of the rectangular window by multiplying discrete data by a “Hanning” window-like function

low-pass filter

keeps all low frequencies, removes high frequencies

high-pass filter

keep all high frequencies, removes low frequencies

band-pass filter

allows a specific range of frequencies to pass through while blocking frequencies both above and below that range