Sorting on datagrid column with binded data and converter

I'm trying to sort data in datagrid, but when I click on the header of the column which has binding with the converter, nothing happens. I use MVVM pattern. Example is attached below. In the example, the grid shows column (Type) which displays type of person and therefore I use converter (class TypeValueConverter). When I use this converter, the grid doesn't sort column Type.

<Window x:Class="GridSort.MainWindow"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        xmlns:my="clr-namespace:GridSort"
        Title="MainWindow" Height="350" Width="525">
    <Grid>
        <DataGrid ItemsSource="{Binding People}" AutoGenerateColumns="False">
            <DataGrid.Resources>
                <my:TypeValueConverter x:Key="typeConverter" />
            </DataGrid.Resources>
            <DataGrid.Columns>
                <DataGridTextColumn Binding="{Binding FirstName}" Header="FirstName" />
                <DataGridTextColumn Binding="{Binding Surname}" Header="Surname" />
                <DataGridTextColumn Binding="{Binding Converter={StaticResource ResourceKey=typeConverter}}" Header="Type" />
            </DataGrid.Columns>
        </DataGrid>
    </Grid>
</Window>

public class ViewModel
{
    private ICollection<Person> people;
    public ICollection<Person> People
    {
        get
        {
            if (this.people == null)
            {
                this.people = new List<Person>();
                this.people.Add(new Student() { FirstName = "Charles", Surname = "Simons" });
                this.people.Add(new Student() { FirstName = "Jake", Surname = "Baron" });
                this.people.Add(new Teacher() { FirstName = "John", Surname = "Jackson" });
                this.people.Add(new Student() { FirstName = "Patricia", Surname = "Phillips" });
                this.people.Add(new Student() { FirstName = "Martin", Surname = "Weber" });
            }

            return this.people;
        }
    }
}

public class TypeValueConverter : IValueConverter
    {
        public object Convert(object value, Type targetType, object parameter, System.Globalization.CultureInfo culture)
        {
            if (value == null)
            {
                return DependencyProperty.UnsetValue;
            }

            return value.GetType().Name;
        }

        public object ConvertBack(object value, Type targetType, object parameter, System.Globalization.CultureInfo culture)
        {
            throw new NotImplementedException();
        }
    }

public abstract class Person
    {
        public string FirstName
        {
            get;
            set;
        }

        public string Surname
        {
            get;
            set;
        }
    }

    public class Student : Person
    {
    }

    public class Teacher : Person
    {
    }

I resolved this issue myself :)

I created new behavior for grid with attached property UseBindingToSort. When I set this property to true then event Sorting on grid is subscribed. After grid fires event sorting I use custom comparer with IValueConverter which is defined in binding. Solution is below:

Change on view

<DataGrid ItemsSource="{Binding People}" AutoGenerateColumns="False"  my:GridSortingBehavior.UseBindingToSort="True">

New behavior with attached property:

public static class GridSortingBehavior
    {
        public static readonly DependencyProperty UseBindingToSortProperty = DependencyProperty.RegisterAttached("UseBindingToSort", typeof(bool), typeof(GridSortingBehavior), new PropertyMetadata(new PropertyChangedCallback(GridSortPropertyChanged)));

        public static void SetUseBindingToSort(DependencyObject element, bool value)
        {
            element.SetValue(UseBindingToSortProperty, value);
        }

        private static void GridSortPropertyChanged(DependencyObject elem, DependencyPropertyChangedEventArgs e)
        {
            DataGrid grid = elem as DataGrid;
            if (grid != null){
                if ((bool)e.NewValue)
                {
                    grid.Sorting += new DataGridSortingEventHandler(grid_Sorting);
                }
                else
                {
                    grid.Sorting -= new DataGridSortingEventHandler(grid_Sorting);
                }
            }
        }

        static void grid_Sorting(object sender, DataGridSortingEventArgs e)
        {
            DataGridTextColumn clm = e.Column as DataGridTextColumn;
            if (clm != null)
            {
                DataGrid grid = ((DataGrid)sender);
                IValueConverter converter = null;
                if (clm.Binding != null)
                {
                    Binding binding = clm.Binding as Binding;
                    if (binding.Converter != null)
                    {
                        converter = binding.Converter;
                    }
                }

                if (converter != null)
                {
                    e.Handled = true;
                    ListSortDirection direction = (clm.SortDirection != ListSortDirection.Ascending) ? ListSortDirection.Ascending : ListSortDirection.Descending;
                    clm.SortDirection = direction;
                    ListCollectionView lcv = (ListCollectionView)CollectionViewSource.GetDefaultView(grid.ItemsSource);
                    lcv.CustomSort = new ComparerWithComparer(converter, direction);
                }
            }
        }

And finally my custom comparer:

class ComparerWithComparer : IComparer
    {
        private System.Windows.Data.IValueConverter converter;
        private System.ComponentModel.ListSortDirection direction;


        public ComparerWithComparer(System.Windows.Data.IValueConverter converter, System.ComponentModel.ListSortDirection direction)
        {
            this.converter = converter;
            this.direction = direction;
        }

        public int Compare(object x, object y)
        {
            object transx = this.converter.Convert(x, typeof(string), null, System.Threading.Thread.CurrentThread.CurrentCulture);
            object transy = this.converter.Convert(y, typeof(string), null, System.Threading.Thread.CurrentThread.CurrentCulture);
            if (direction== System.ComponentModel.ListSortDirection.Ascending){
                return Comparer.Default.Compare(transx, transy);
            }
            else
            {
                return Comparer.Default.Compare(transx, transy) * (-1);
            }
        }
    }

And that's all.