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

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.

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

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.

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.

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

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)

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.

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

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

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).

Sampling Aliasing I – The Movie (FS=200, Range 50-150 Hz) (click on figure to run or pause)

Sampling Aliasing II – The Cheap Sequel (FS=200, Range 350-450 Hz) (click on figure to run or pause)

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.

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:

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

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

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)

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

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