1. Sampling and Quantization
EE422G Signals and Systems Laboratory
Sampling and Quantization
Kevin D. Donohue Electrical and Computer Engineering University of Kentucky

2. Sampling Function
A link between the analog and discrete worlds can be analyzed using a train of Dirac Delta functions:
where the impulses are separated by T seconds (sampling interval), which corresponds to a sampling frequency of:
𝑠 𝑡 = 𝑛=−∞ ∞ δ 𝑡−𝑛𝑇
Ω 𝑠 =2π 𝐹 𝑠 = 2π 𝑇

3. FT of Sampling Function
Recall from the FS review, that the FS of this train of impulse functions is:
The Fourier Transform (FT) of s(t) can then be written as:
𝑠 𝑡 = 𝑘=−∞ ∞ 1 𝑇 exp 𝑗2π𝑘 𝐹 𝑠 𝑡
𝑆 ˆ 𝑓 = 𝑘=−∞ ∞ 1 𝑇 δ 𝑓−𝑘 𝐹 𝑠

4. Bandlimited Signal
Consider a bandlimited signal xc(t) with spectrum shown below:
Note that all frequency content is less than FN.
If xc(t) is multiplied by the sampling function s(t) to isolate discrete values, the multiplication corresponds to convolution in the frequency domain:
𝑠 𝑡 𝑥 𝑐 𝑡 ⇔𝑆 𝑓 ∗ 𝑋 𝑐 𝑓

5. Aliasing
Aliasing is the result of convolving the signal spectrum by the sampling function spectrum:
Example: Aliased spectra when FN < Fs/2 (Fs/2 is referred to as the Nyquist frequency)
𝑋 𝑠 ˆ 𝑓 = 𝑆 ˆ 𝑓 ∗ 𝑋 𝑐 ˆ 𝑓 = −∞ ∞ 𝑘=−∞ ∞ 1 𝑇 δ 𝑓−λ−𝑘 𝐹 𝑠 𝑋 𝑐 ˆ λ 𝑑λ
𝑋 𝑠 ˆ 𝑓 = 𝑘=−∞ ∞ 1 𝑇 𝑋 𝑐 ˆ 𝑓−𝑘 𝐹 𝑠

6. Reconstructing Analog Signal
The original analog signal in theory can be reconstructed from its samples without error as long as Fs > 2FN using an ideal low-pass filter to filter out the non-interfering aliased spectra, the original continuous time signals can be recovered without distortion
Ideal low-pass filter with cut-off frequency greater than FN and less than FS - FN

7. Sampling Above and Below Nyquist Rate
Sampling with Nyquist Frequency greater than FN (Fs > 2FN)
Sampling with Nyquist Frequency below FN (Fs < 2FN ), where distortion arises due to overlapping spectra, the signal cannot be recovered without error using low-pass filtering.

8. Aliasing Example
Given sampled at Fs = 8kHz. Find the frequency after sampling aliased to the range 0 to Fs/2.
Note FT of is
which actually consists of 2 frequencies
Now apply aliasing formula:
Find n and f so that arguments of the Dirac deltas are 0 and f is between
For n = 2 (using appropriate sign for each Dirac delta argument), f =  Hz, so aliased frequency is 1000 Hz
𝑥 𝑡 =3cos 2π15k𝑡
𝑋 ˆ 𝑓 =𝐴 δ 𝑓− 𝑓 0 +δ 𝑓+ 𝑓 0 2
𝑥 𝑡 =𝐴cos 2π 𝑓 0 𝑡
± 𝑓 0
𝑋 𝑠 ˆ 𝑓 = 𝑛=−∞ ∞ 1 𝑇 𝑋 ˆ 𝑓−𝑛 𝐹 𝑠
± 𝐹 𝑠 2
𝑋 𝑠 ˆ 𝑓 = 𝑛=−∞ ∞ δ 𝑓−𝑛8k−15k +δ 𝑓−𝑛8k+15k

9. Aliasing The Movie I
The following movie superimposes an input sinusoid in red with its output after sampling in blue. The sampling frequency is 200 Hz and the input frequency ranges from 50 Hz to 150 Hz. If program will not play movie in browser, you can download avi movie file from

10. Aliasing The Movie II
The following movie superimposes an input sinusoid in red with its output after sampling in blue. The sampling frequency is 200 Hz and the input frequency ranges from 350 Hz to 450 Hz. If program will not play movie in browser, you can download avi movie file from

11. Flat Top Sampling
Practical sampling devices do not operate as delta functions, but more like short duty cycle pulses where the signal is averaged over the interval to create the sample. The sampling function for this case becomes:
where the rect() function is zero everywhere except for t between -TP/2 and TP/2, where it equals 1, and TP  T. So this impacts the previous aliasing formula for signal x(t) in the following way (convolution in time corresponds to multiplication in frequency):
𝑠 𝑝 𝑡 =rect  𝑡 𝑇 𝑝  ∗𝑠 𝑡 = 𝑛=−∞ ∞ rect  𝑡−𝑛𝑇 𝑇 𝑝 
𝑋 𝑠 ˆ 𝑓 =sinc π𝑓 𝑇 𝑝  𝑘=−∞ ∞ 1 𝑇 𝑋 𝑐 ˆ 𝑓−𝑘 𝐹 𝑠

12. Flat Top Sampling
Demonstration of how flattop sampling distorts spectrum even in fundamental range. For smaller duty cycles relative to sampling period the distortion is minimized. Sampling frequency in all examples is 100 Hz with integration intervals of 1, and 0.01 times the sampling period (0.01 seconds).