EE513 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 interactions. If x(t) is the time domain signal, its Laplace transform is defined as: where s belongs to the set of complex numbers over which the integral converges. What is the Laplace transform of a system impulse response? What does the Laplace transform become if it is evaluated along the imaginary axis?

Z-Transform If the Laplace transform is made discrete by sampling the time axis with interval T s, it becomes: Now let:

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

j  Axis in s to z Mapping j  axis in s corresponds to a zero real part,  = 0 for all imaginary values, . In z this results in: S-plane IM RE Z-plane IM

Z-plane S-plane Relationships Imaginary axis (j  ) in S-plane, maps into the unit circle in Z- plane, where segments on the j  axis described by: map into a complete unit circle for every integer k. Since the mapping from S to Z is not unique, let the range for k=0 be referred to as the root frequency range and all other values of k with the aliased frequency ranges.

Negative Real Axis s to z Mapping Negative real axis in s corresponds to  < 0. In z this results in: Z-plane RE IM S-plane IM RE

Z-plane S-plane Relationships The negative real axis of the S-plane (  < 0 ) maps into the area inside the unit circle of Z (| z | < 1). Therefore the stable region of the S-plane (left-half plane) corresponds to the area inside the unit circle of the Z-plane. Similarly, the unstable region of the S-plane (right-half plane) corresponds to the area outside the unit circle of the Z- plane. Because of aliasing from the sampling and scaling from the exponential transformation, there is no simple (linear) scaling between the Z and S planes. A warping or distortion occurs when matching domain points: Warping Possible Aliasing

Z-Transform One-Sided Many applications assume the input starts at t = 0 (n=0 for discrete) and no response exists before t = 0. So the Z-transform is often written as: Examples: Find the z-transforms of x[n] = u[n] and x[n] = a n u[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) Use definition for k > 0. Use definition Use ZT properties (delay)

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.

Sinusoidal Response Consider sinusoidal input: Note this input is always on (steady-state). Show that for impulse response h(n), the convolution sum for evaluating the output becomes: Important Concepts:  The response to a sinusoidal input is a phasor multiplication between input phasor and transfer function value at the excitation frequency.  The frequency response (Transfer Function) of a discrete system is the z- transform of its impulse response evaluated on the unit circle!  Convolution in time domain is equivalent to multiplication in the frequency domain. Input Complex Coefficient

Sinusoidal Response Example For sinusoidal input: And system described by: Derive an expression for and plot the frequency response (phase and magnitude) of the system 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 3 samples (2 complete sampling intervals) must fall within a period for digitization without aliasing. In other words the sampling rate must be greater than twice the highest signal frequency for a band limited signal. F s =200 Hz T s = 5 ms

Sampling Aliasing I – The Movie (FS=200, Range Hz) Run following mfile at:

Sampling Aliasing II – The Sequel (FS=200, Range Hz). Change beginning and ending frequency parameters in following mfile to run:

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

Aliased Signal Spectra Sampling in time with sampling frequency f s creates an infinite pattern of a shifted analog spectra so that frequency domain has a periodicity of f s. Let S a (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) a) Compute impulse response directly by hand b) Use Matlab function “filter” c) Take inverse of Z transform d) Examine poles and zeros of filters

Linear Difference Equations (FIR)

Linear Difference Equations (IIR)

Homework (4) Find the impulse response of the following filters. a) b) c) 1) Use Matlab function “filter” 2) Take inverse of Z transform 3) Examine poles of filter and comment on expected stability