1. EE599-020 Audio Signals and Systems
Digital Signal Processing (Systems)
Kevin D. Donohue Electrical and Computer Engineering University of Kentucky

2. Laplace to begin with
The Laplace transform is used to characterize and analyze signal and system interaction. If x(t) is the time domain signal, its Laplace transform is defined as:
where s belongs to the set of complex numbers such that the integral converges. Describe ways that system characteristics are derived from the Laplace transform of its impulse response.

3. Z-Transform
If the Laplace transform is made discrete by sampling the time axis with interval , it becomes:
Let

4. Z-Transform Substitute z to obtain:
Normalize sampling rate and define Z-transform as:
where z belongs to the set of complex numbers such that the summation exists. Describe mapping from S-plane to Z-plane.

5. Z-Transform One-Sided
Many applications assume the input starts at t = 0 and there is no response before t = 0. So the Z-transform is often written as:
Examples find the z-transforms of x(n) = u(n); x(n) = anu(n); Assuming the z-transform of x(n) is X(z), find the Z-transform of x(n-k) for k>0.

6. Homework(1)
Find the z-Transforms of: a) b) c)

7. Convolution
Given the impulse response of a discrete linear system h(n) the input-output relationship is described by discrete convolution:
For x(n) and h(n) below, graphically demonstrate their convolution.
x(n)
h(n)

8. Sinusoidal Response Consider sinusoidal input:
Show that for impulse response h(n), the output can be expressed via convolution as:
Important Concepts: The frequency response of a discrete system is its z-transform evaluated on the unit circle! So substitute
and evaluate for Also … Convolution in time is multiplication in the Z-domain.

9. Sinusoidal Response Example
For sinusoidal input:
And system described by:
Plot the frequency response (phase and magnitude) and compute the output.

10. Homework(2) For sinusoidal inputs: And system described by:
Plot the frequency response (phase and magnitude) and compute the corresponding outputs.

11. Sampling
Sampling rate determines the highest signal frequency that can be reconstructed from the signal samples without error. At least 2 samples must fall within a half cycle for it to be digitized without aliasing (sampling rate must be greater than 2 twice the highest signal frequency).

12. Sampling
Aliasing I – The Movie (FS=200, Range Hz) (click on figure to run or pause)

13. Sampling
Aliasing II – The Cheap Sequel (FS=200, Range Hz) (click on figure to run or pause)

14. The Sampling Theorem
A band-limited continuous signal s(t) can be reconstructed without error from its samples provided:
where fs is the sampling frequency in samples per second, and fb is the frequency above which s(t) has no energy.

15. Aliased Signal Spectra
Sampling in time with sampling frequency fs creates a shifting of the analog spectra that results in a frequency domain periodicity of fs. Let Sa(f) be the spectra of the original analog signal, the spectrum of the sampled signal becomes:

16. Aliased Signal Spectra
Spectral periodicity of a low-pass signal (not really band-limited) resulting from an 8 kHz sampling

17. Restoring Sampled Signals
A sampled signal is reconstructed by low-pass filtering the samples with a cut-off near the folding frequency

18. Aliased Signal Example
Before sampling at a given rate, signals are often low-pass filtered (anti-aliasing filter) to limit distortions from aliasing.
Original Sound
Limited Bandwidth (LPF with 900 Hz cutoff) and sampled at 2 kHz
Original Sound sampled at 2 kHz (aliasing)

19. Homework(3) Determine the aliased frequencies in the range of
For the following sampling frequency and signal pairs:

20. Linear Difference Equations
Discrete systems are described by Z-transforms in the frequency domain and difference equations in the time domain. Digital filters can be designed in either domain.
Find the impulse response of the following filters.
(FIR)
(IIR)
Compute impulse response directly by hand
Use Matlab function “filter”
Take inverse of Z transform
Examine poles and zeros of filters